32 FA15 Abstracts

IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed.

For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme. rial on Yang-Baxter graphs and Macdonald polynomials is In particular, for superintegrable systems in 3D, the poly- based on joint work with J.-G. Luque. nomial representations of quadratic algebras are given in terms of two-variable polynomials and the two-variable Charles F. Dunkl analog of the Askey scheme, including the quadratic Racah University of Virginia algebra, will be discussed along with the limiting processes [email protected] within the scheme.

Sarah Post IP2 University of Hawaii at Manoa Two Variable q-Polynomials Department of Mathematics [email protected] Abstract not available at this time.

Mourad Ismail IP7 University of Central Florida A New Look at Classical Orthogonal Polynomials [email protected] There are two possible definitions of classical orthogo- nal polynomials:(i) they satisfy a second order differential IP3 or difference equation;(ii) (generalized) derivative of them On the Asymptotic Behavior of a Log Gas in the gives again orthogonal polynomials. Both definitions are Bulk Scaling Limit in the Presence of a Varying related with concrete forms of corresponding operators. We External Potential propose a new approach dealing with some abstract um- bral operators. This gives a wide generalization of a notion Abstract not available at time of publication. of classical orthogonal polynomials. Percy Deift Alexei Zhedanov Courant Institute Donetsk Institute for Physics and Engineering, Ukraine New York University [email protected] [email protected]

IP4 IP8 Title Not Available at Time of Publication Asymptotic and Numerical Aspects of Special Functions Abstract not available at this time. For the numerical evaluation of special functions, asymp- Olga Holtz totic expansions are an important tool. The standard ex- University of California, Berkeley pansions can be used rather straightforwardly. The so- Technische Universitaet Berlin called uniform expansions need more attention, especially [email protected] for critical values of secondary parameters in the asymp- totic problem. For example, the Airy-type expansion of the Bessel function Jν (z) can be used for large domains of the IP5 argument and order, but for the transition value z = ν spe- Integrable Probability and the Role of Painlev cial methods are needed for computing the coefficients. We Functions mention several methods for handling this type of problem. We start with a few examples for which Maple and Mathe- We will review various models in probability that are inte- matica have problems in the evaluation of well-known spe- grable in the sense that various distribution functions can cial functions, like the Kummer U-function, for medium- be explicitly evaluated in terms of Painleve functions and sized values of the parameters. We discuss recent activities their generalizations. We develop in more detail a class in the Santander-Amsterdam project on the evaluation of of stochastic growth models that belong to the Kardar- special functions, in particular for certain cumulative dis- Parisis-Zhang (KPZ) uni- versality class such as the asym- tribution functions. We start with the incomplete gamma FA15 Abstracts 33

functions, and we give recent results for the non central chi- polynomials, which are multivariate orthogonal polynomi- squared or the non central gamma distribution, also called als. Recently Eric Rains defined moments of Koornwinder Marcum Q−function in radar detection and communica- polynomials at q=t, which appear to be polynomials with tion problems. This is joint work with Amparo Gil and positive coefficients when written appropriately in the pa- Javier Segura (University of Cantabria, Santander, Spain). rameters of the ASEP. I’ll explain joint work with Sylvie Corteel in which we show that Koornwinder moments at q=t are related to the 2-species ASEP, an exclusion process Nico M. Temme involving two different types of particles. I’ll also describe Centrum Wiskunde & Informatica complementary work of Olga Mandelshtam and Xavier Vi- The Netherlands ennot providing a combinatorial description of the station- [email protected] ary distribution of the 2-species ASEP.

Lauren Williams IP9 University of California Berkeley Multivariate Orthogonal Polynomials and Modified [email protected] Moment Functionals

Multivariate orthogonal polynomials can be defined by SP1 d means a measure defined on a domain on R . A very impor- SIAG/OPSF Gbor Szeg Prize Announcement and tant class of multivariate orthogonal polynomials is called Lecture classical because the measure satisfies a matrix analogue of the Pearson differential equation as well as the orthogo- Abstract not available at time of publication.. nal polynomials are the eigenfunctions of a partial second order differential operator. In this talk, we present old and Karl Liechty new results on classical multivariate orthogonal polynomi- DePaul University als. In particular, some classical multivariate orthogonal Department of Mathematical Sciences polynomials and some useful modifications will be studied, [email protected] as well as their impact into the useful properties of the orthogonal polynomials. We study the so–called Uvarov modification obtained by adding to the measure one or a CP1 finite set of mass points. Recently, Christoffel modifica- Killip-Simon Problem and Jacobi Flow on Gsmp tion in several variables, that is, the modification obtained Matrices by multiplying the measure times a polynomial, has been studied in the frame of linear relations. D. Damanik, R. Killip and B. Simon completely described the spectral properties of Jacobi matrices J+,whicharein 2 Teresa E. P´erez asence perturbations of the isospectral torus of periodic Universidad de Granada Jacobi matrices. The spectrum of a periodic Jacobi matrix Departamento de Matem´atica Aplicada is a system of intervals of a very specific nature. Jointly [email protected] with P. Yuditskii, we generalize this result to spectral sets, which are arbitrary finite system of intervals.

IP10 Benjamin Eichinger Hypergeometric Series: On Number Theory’s Se- Institute of Analysis, Johannes Kepler University Linz cret Service [email protected]

A natural outcome of the theory of generalized hypergeo- CP1 metric functions are rational approximations to the values of Riemann’s zeta functions and alike mathematical con- A Matrix Approach for the Semiclassical and Co- stants. In my talk I plan to outline the way it goes (hyper- herent Orthogonal Polynomials geometric series, hypergeometric Barnes- and Euler-type integrals) and stress on some recent achievements — the We obtain a matrix characterization of semiclassical or- current best irrationality measures of π (due to Salikhov), thogonal polynomials in terms of the Jacobi matrix as- of log 2 (due to Marcovecchio) and of ζ(2). The final part sociated with the multiplication operator in the basis of of the talk will discuss some linear and algebraic indepen- orthogonal polynomials, and the lower triangular matrix dence results that make use of generalized hypergeometric that represents the orthogonal polynomials in terms of the functions. monomial basis of polynomials. We also provide a matrix characterization for coherent pairs of linear functionals. Wadim Zudilin University of Newcastle Lino G. Garza [email protected] Universidad Carlos III de Madrid [email protected]

IP11 Francisco Marcell´an Orthogonal Polynomials and the 2-Species ASEP Universidad Carlos III de Madrid Instituto de Ciencias Matem´aticas (ICMAT) The asymmetric exclusion process (ASEP) is a model of [email protected] particles hopping on a 1-dimensional lattice with open boundaries. The partition function of this model is related Luis E. Garza to moments of Askey-Wilson polynomials. Askey-Wilson Departamento Matematicas,Universidad Nacional polynomials are at the top of the hierarchy of orthogonal Colombia polynomials, and are also a limiting case of Koornwinder [email protected] 34 FA15 Abstracts

Natalia C. Pinz´on-Cort´es [email protected] Universidad Nacional de Colombia [email protected] CP2 Approximate Controllability of An Impulsive Neu- CP1 tral Differential Equation with Deviating Argu- An Explicit Family of Askey-Wilson Type Matrix- ment and Bounded Delay Valued Orthogonal Polynomials In this paper we prove the approximate controllability of an We discuss an explicit set of matrix-valued orthogonal impulsive neutral differential equation with deviated argu- polynomials, which are eigenfunctions to a matrix-valued ment and control parameter included in the nonlinear term. Askey-Wilson type q-difference operator. A study of the We use Schuader fixed point theorem and fundamental as- corresponding matrix-valued weight leads to an expres- sumptions on system operators to prove the result; thereby sion of the matrix entries of these polynomials in terms removing the need to assume the invertibility of a control- of polynomials from the q-Askey-scheme. Other properties lability operator which fails to exist in infinite dimensional of these polynomials as well as an outlook will be discussed. space if the generated semigroup is compact. We also give (Joint work with Noud Aldenhoven and Pablo Rom´an.) an example to illustrate our result. In this paper we study the approximate controllability of the following problem Erik Koelink, Noud Aldenhoven d[x(t)+g(t, xt)] Radboud Universiteit = A[x(t)+g(t, xt))] + Bu(t)+f(t, x(a(x(t),t)),u(t)), [email protected], [email protected] dt x0(θ)=φ(θ),θ ∈ [−r, 0] + − Pablo Rom´an x(tk ) − x(tk )=Ik(x(tk)),k=1, ..., m, Univesidad Nacional de C´ordoba [email protected] .

Sanjukta Das CP1 Indian Institute of Technology, Roorkee, Jacobi Polynomial Moments and Products of Ran- sanjukta [email protected] dom Matrices Dwijendra Pandey In this talk we study global singular value distributions Indian Institute of Technology, Roorkee arising from products of complex Gaussian random matri- [email protected] ces and truncations of Haar distributed unitary matrices. To this end, we introduce an appropriate general class of measures with moments essentially given by specific Jacobi CP2 polynomials with varying parameters. Solving the under- On the Study of Solution of Lorenz System Using lying moment problem is based on a study of the Riemann Generalized Mittag-Leffler Function surfaces associated to a class of algebraic equations. In this talk, generalized Mittag-Leffler function of Prab- Thorsten Neuschel hakar (1971) and Shukla et al. (2009) are applied to solve Department of Mathematics, KU Leuven approximate and analytical solutions of nonlinear frac- Leuven, Belgium tional differential equation system such as Lorenz system. [email protected] The obtained results were compared with the results of Homotopy perturbation method and Variational iteration method. CP2 Ranjan K. Jana Special Functions As Non-Uniform Steady-State S. V. NATIONAL INSTITUTE OF TECHNOLOGY Solutions of a Mathematical Model Describing (SVNIT) Fluid-Solute Transport in Peritoneal Dialysis SURAT, GUJARAT, INDIA [email protected] Recently a mathematical model for fluid and solute trans- port in peritoneal dialysis has been proposed (R. Cherniha et al, Int. J. Appl. Math. Comp. Sci., 2014, V.24, P.837- CP2 851). The model is based on a nonlinear system of partial Hypergeometric Functions and Chebyshev Polyno- differential equations for fluid and solute concentrations. mials: Explicit Solutions of 2-D Free Boundary The model was simplified in order to obtain exact formulae Problems in Groundwater Hydrology for steady-state solutions, hence, exact solutions involving Bessel and hypergeometric functions for the fluid and so- Steady, phreatic, Darcian flows in hillslope hydrology [1] lute fluxes are constructed. Biomedical interpretation of are generalized to bedrocks with faults, karst or zonal frac- mathematical results is provided. turing. By conformal mappings and solution of Hilberts boundary value problems, characteristic functions (deriva- Roman Cherniha tives of complex potential/coordinate) are expressed as School of Mathematical Sci. University of Nottingham combinations of hypergeometric functions with three over- [email protected] lapping convergence domains. If a part of impermeable substratum is a control variable the kernel of the Cauchy- Jacek Waniewski type integral is expanded into a series of Chebyshev poly- IBIB of Polish Acad. Sci. nomials. Accuracy of integrations and series truncations is Warsaw, Poland studied. 1. Kacimov A.R., Obnosov Yu.V., Abdalla, O., FA15 Abstracts 35

Castro-Orgaz O. (2014) Groundwater flow in hillslopes: an- conjugated Dirac operators, is considered. The reproduc- alytical solutions by the theory of holomorphic functions ing kernel for this space is obtained and expressed in terms and hydraulic theory. Applied Mathematical Modelling, of Jacobi polynomials. doi:10.1016/j.apm.2014.11.016. Michael Wutzig, Hendrik De Bie, Frank Sommen Anvar Kacimov Ghent University Sultan Qaboos University [email protected], [email protected], [email protected] [email protected]

Yurii Obnosov Kazan Federal University, Russia CP3 Institute of Mathematics and Mechanics New Index Transforms with the Product of Bessel [email protected] Functions. In this talk we discuss new index transforms with the prod- CP3 uct of Bessel functions as kernels. In particular, we will investigate a generalization of the Lebedev transform with Approximation by a Generalization of the the square of the Macdonald function. Mapping properties Jakimovski-Leviatan Operators are studied, inversion formulas are proved. As applications, We introduce a generalization of the Jakimovski–Leviatan- initial value problems for higher order partial differential Kantorovich Type operators constructed by A. Jakimovski and difference equations are solved. and D. Leviatan (1969) and the theorems on convergence Semyon Yakubovich and the degree of convergence are established. Further- University of Porto more, we study the convergence of these operators in a Faculty of Sciences, Dept. of Mathematics weighted space of functions on a positive semi-axis. [email protected] Didem Aydin Ari Kirikkale University CP4 Kirikkale/ Turkey [email protected] Representations for the Parameter Derivatives of Orthogonal Polynomials in Two Variables

Sevilay Kirci Serenbay The aim of this paper is to obtain the parameter derivative Baskent University representations in the form of Ankara / Turkey [email protected] n−1 m k ∂Pn,k(λ; x, y) = dn,j,mPm,j (λ; x, y)+ en,j,kPn,j (λ; x, y) ∂λ m=0 j=0 j=0 CP3 Integrals Involving Powers of K for some families of orthogonal polynomials in two variables with respect to their parameters by using the parameter (α,β) Let K denote the complete elliptic integral of the first kind. derivatives of the classical Jacobi polynomials Pn (x) We investigate integrals of the form where λ is a parameter and 0 ≤ k ≤ n; n, k =0, 1, 2, ....  1 Furthermore, the orthogonality properties of the paramet- n f(x)K(x) dx, ric derivatives of these polynomials are discussed. 0 Rabia Aktas where f is an algebraic function. We show that, for all Ankara University natural n, there exist classes of f such that this integral [email protected] may be evaluated in closed form; e.g.

 1 8 3 3Γ(1/4) CP4 K(x) dx = 2 . 0 1280π Sums of Squared Baskakov Functions

These results are proven using Eisenstein series and Fourier The complete monotonicity of sums of squares of gener- series, and are connected to multiple sums arising from alized Baskakov basis functions is proved by deriving the chemical lattices. corresponding result for certain hypergeometric functions. This implies its logarithmic convexity. Thereby a recent James G. Wan conjecture of Ioan Rasa is established. Moreover in the Singapore University of Technology and Design central Baskakov case the limit distribution of the zeros is james [email protected] computed for large values of a parameter. As a byproduct new representations for the complete elliptic integral of the first kind are established. CP3 Reproducing Kernels of Spherical Monogenics Wolfgang Gawronski Department of Mathematics On the space of spherical harmonics there exists a repro- University of Trier ducing kernel that is given by a Gegenbauer polynomial. [email protected] By going over to complex variables, one obtains a reproduc- ing kernel expressed as a Jacobi polynomial. In this talk, Ulrich Abel the space of hermitian monogenics, which is the space of Technical University of Applied Sciences Mittelhessen polynomial bihomogeneous null-solutions of two complex Germany 36 FA15 Abstracts

[email protected] visualizations using WebGL, a JavaScript API for render- ing 3D graphics in a web browser without a plugin. Thorsten Neuschel Department of Mathematics, KU Leuven Bonita V. Saunders, Brian Antonishek, Qiming Wang Leuven, Belgium National Institute of Standards and Technology (NIST) [email protected] [email protected], [email protected], [email protected]

CP4 Bruce Miller On the Horizontal Monotonocity of the Gamma Applied and Computational Mathematics Division Function National Institute of Standards and Technology [email protected] The behaviour of the gamma function along vertical lines in the right half plane has been studied closely but very lit- tle on the modulus of the gamma function along horizontal CP5 lines in the upper half plane. We show that |Γ(t + ia)| is Extension of Generalized Mittag-Leffler Density monotone in t as soon as the imaginary part a exceeds a and Processes threshold value. We give a proof of this and indicate some sharp lower bounds for a beyond which this monotonic be- Shukla and Prajapati [J. Math. Anal. Appl. No haviour holds. 336(2007),797-811] extended the Mittag-Leffler function γ,q ∞ (γ)qn · zn and defined as Eα,β(z)= n=1 Γ(αn+β) n! ,where Gopala Krishna Srinivasan α, β, γ ∈ C, (α) > 0, (β) > 0, (γ) > 0,q ∈ (0, 1) ∪ N, Indian Institute of Technology Bombay γ,q and the function Eα,β(z) converges absolutely for all z ∈ C [email protected] −1 if q<(α) + 1 (entire function of order (α) )andfor |z| < 1ifq = (α)+1. In this paper, we obtain some prop- CP5 erties of Mittag-Leffler density, Structural representation of the Mittag-Leffler variable and Mittag-Leffler Stochas- γ,q Orthogonal Polynomial Interpretation of Δ-Toda tic Process by using E (z). and Volterra Lattices α,β Pratik V. Shah The correspondence between dynamics of Δ-Toda and Δ- C.K.Pithawalla College of Engg. & Technology, Volterra lattices for the coefficients of the Jacobi opera- Surat - 395007 , Gujarat , INDIA. tor and its resolvent function is established. A method [email protected] to solve inverse problem –integration of Δ–Toda and Δ- Volterra lattices – based on Pad´e approximants and con- tinued fractions for the resolvent function is proposed. The Ajay Shukla main ingredient are orthogonal polynomials which satisfy S.V.National Institute of Technology, an Appell condition, with respect to the forward differ- Surat 395007, Gujarat, India. ence operator Δ. It is shown that the Δ-Volterra lattice is ajayshukla2@rediffmail.com related to the Δ-Toda lattice by Miura or B¨acklund trans- formations. CP5 Ana Foulquie Associated Polynomials, Markov’s Theorem, and Universidade de Aveiro First-Hitting Times of Birth-Death Processes [email protected] A classical result of Karlin and McGregor states that for a birth-death process on the nonnegative integers with corre- Ivan Area sponding birth-death polynomials {Qn}, the Laplace trans- Universidad de Vigo form of the first-hitting time Tij satisfies [email protected]

sTij Qi(s) Amilcar Branquinho E[e ]= ,s<0, Qj (s) Universidade de Coimbra [email protected] provided ij, featuring associated polynomials of {Qn}.It [email protected] will be shown in the talk that these results may also be obtained with the help of Markov’s theorem.

CP5 Erik A. Van Doorn State of the Art Visualizations of Complex Func- University of Twente tion Data [email protected]

The NIST DLMF contains function visualizations that would have been difficult, if not impossible, to envision CP6 when Abramowitz and Stegun was published. We have Ring of Integrals Operators improved quality and accessibility using mesh generation, approximation theory, and the latest 3D web graphics tech- we define an ring of linear operators whose kernels are es- nology to capture key features such as zeros, poles, and sentially Bessel functions. These operators act on spaces of branch cuts. We demonstrate our redesign of the function entire functions of exponential type and are endowed with FA15 Abstracts 37

a complete and explicit symbolic calculus that we will de- [email protected] scribed. We introduce a new symbols to obtain a new class polynomials. CP7 Miloud Assal A Q-Extension of a Reduction Formula of Watson Institut Sup´erieur des Math´ematiques Appliqu´ees [email protected] In this paper, we give a reduction formula for the q-integral  1   α n γ β+1 In = x (qx; q)β Dq−1 x (q x; q)δ dqx, n =0, 1, 2,.. CP6 0 Stable Regions of Tur´an Expressions where α, β, γ, and δ are complex numbers. The reduc- tions formula is a three term recurrence relations of In. Consider polynomial sequences that satisfy a first-order dif- The representation of In in terms of basic hypergeometric ferential recurrence. We prove that if the recurrence is series will be investigated. This is a q-analogue of the work of a special form, then the Tur´an expressions for the se- introduced by W. N. Bailey in 1953. quence are weakly Hurwitz stable (non-zero in the open right half-plane). A special case of our theorem settles Zeinab Mansour a problem proposed by S. Fisk that the Tur´an expressions King Saud University Facuty of Science, Mathematics for the univariate Bell polynomials are weakly Hurwitz sta- Dep. ble. We obtain related results for Chebyshev and Hermite Riyadh P.O. Box 2455 Riyadh 11451 Saudi Arabia polynomials, and propose several extensions involving La- [email protected] guerre polynomials, Bessel polynomials, and Jensen poly- nomials associated to a class of real entire functions. This Maryam AL Towaileb is joint work with Lukasz Grabarek (University of Alaska) Kingdom of Saudi Arabia and Mirk´o Visontai (Google Reseach). P.O.Box 22459, Riyadh 11495 [email protected] Matthew Chasse Rochester Institute of Technology CP7 [email protected] Some Q-Continued Fractions and Their Connec- tions with Lambert Series and Mock Theta Func- CP6 tions New Characterizations of Leonard Pairs Mock theta function is last gift to Mathematical world by S.Ramanujan,that he quoted in his last letter to G.H. Leonard pairs are pairs of linear transformations that act Hardy on January 12,1920. Lambert series is a well known on each other’s eigenspaces in an irreducible tridiagonal series used by Ramanujan to prove many of his identities. fashion. They are related to the Askey scheme of orthogo- Following Ramanujan, many mathematicians like George nal polynomials, distance-regular graphs, and the represen- Andrews and B.C Berndt also used these series to prove tation theory of Lie algebras. In this talk, we will discuss many of Ramanujan identities. In the present work, by two new characterizations of Leonard pairs using certain making use of Andrews and Hickerson identities, we have sequences of parameters. established certain new q-continued fractions for the ratio of Lambert series and also for the ratio of combination of mock theta function of order six. Edward Hanson State University of New York at New Paltz Mohan Rudravarapu [email protected] Government Polytechnic,Srikakulam Andhra Pradesh,India [email protected] CP6 A Classification of the Lowering-Raising Triples. Pankaj Srivastava Motilal Nehru National Instituite of Technology,Allahabad Let F denote a field, and let V denote a vector space over [email protected] F with positive dimension d + 1. By a decomposition of V we mean a sequence of one-dimensional subspaces whose { }d direct sum is V .Let Vi i=0 denote a decomposition of CP8 V . A linear transformation A ∈ End(V )issaidtolower d On Verblunsky Coefficients Related to a Particular {Vi}i=0 whenever AVi = Vi−1 for 1 ≤ i ≤ d and AV0 =0. d Class of Carath´eodary Functions The map A is said to raise {Vi}i=0 whenever AVi = Vi+1 for 0 ≤ i ≤ d − 1andAVd = 0. A pair of elements A, B in Schur algorithm provides a procedure to obtain Szeg¨opoly- End(V ) is called lowering-raising (or LR) whenever there nomials from Carath´eodary functions defined on the unit exists a decomposition of V that is lowered by A and raised disk by means of PC-fractions. The Verblunsky coefficients by B. A triple of elements A, B, C in End(V ) is called LR obtained from these Szeg¨o polynomials are useful in char- whenever any two of A, B, C form an LR pair. We classify acterizing various interesting properties of these polyno- up to isomorphism the LR triples on V .Wediscusshow mials. In this work, a class of Verblunsky coefficients re- LR triples are related to the quantum group Uq (sl2). lated to a particular class of Carath´eodary functions avail- able in the literature are considered. The corresponding Paul M. Terwilliger Chain sequences and minimal parameter sequences related Math Department, University of Wisconsin-Madison to these Verblunsky coefficients are investigated. Using 38 FA15 Abstracts

a well-known procedure, the corresponding para orthogo- [email protected] nal polynomials and Szeg¨o polynomials are reconstructed which lead to interesting observations. Various related ap- plications for these coefficients are obtained and compared CP9 with the results exist in the literature. Generalized Hurwitz Matrices: Criteria of Total Positivity Swaminathan Anbhu I.I.T. Roorkee Given an infinite-dimensional generalized Hurwitz matrix [email protected], [email protected] HM (f)={aMj−i}ij , defined by a polynomial f(x)= n n−1 a0x + a1x + ...+ an, ai ∈ R and a positive integer M ≤ n, we provide a necessary and sufficient criterion for CP8 its total positivity. Our criterion is given in the form of OnZerosofaClassofRealSelf-ReciprocalPoly- a finite number of determinantal conditions. We also con- nomials sider a factorization of a totally positive generalized Hur- witz matrix using a generalization of the classical Euclidean We investigate the distribution, the simplicity and the algorithm. monotonicity properties of the zeros around the unit circle Olga Y. Kushel and real line of the real self-reciprocal polynomials given 2 n−1 n Technische Universitaet Berlin by Rn(z)=1+d(z +z +···+z )+z ,whered is a real [email protected] number and n>0. The sequence of polynomials {Rn} satisfies a three term recurrence relation, this recurrence relation is used to get further results. Olga Holtz University of California-Berkeley Cleonice F. Bracciali [email protected] UNESP - Universidade Estadual Paulista Dept of Applied Mathematics Sergey Khrushchev [email protected] Kazakh-British Technical University svk [email protected] Vanessa Botta, Junior Pereira UNESP - Universidade Estadual Paulista Mikhail Tyaglov [email protected], [email protected] Shanghai Jiao Tong University [email protected]

CP8 CP9 Orthogonal Polynomials with As Their Weight Functions Canonical Vector-Polynomials for Complex Order Modified Bessel Functions Abstract. The orthogonal polynomials are used very often The canonical vector-polynomials for the approximation of in applied mathematics M. Podisuk and his colleague pre- modified Bessel functions of the second kind with com- sented some kinds of orthogonal polynomials in I, II, III plex order and real argument are constructed. Lanczos and IV. In this paper, the orthogonal polynomials in the Tau method computational scheme is applied for the ap- closed interval [0,1] with the weight functions of the form proximation of the solutions of hypergeometric type dif- will be illustrated. ferential equations and their systems. The stable recur- rent scheme is developed for the calculation of the canon- Maitree -. Podisuk ical vector-polynomials coefficients. Some applications for Kasem Bundit University mixed boundary value problems in wedge domains are [email protected] shown.

Juri M. Rappoport CP8 Russian Academy of Sciences Discrete Orthogonal and q-Orthogonal Polynomial [email protected] Sequences in the Extended Hahn Classes with Sim- ple Recurrence Coefficients CP9 In [1] we found explicit formulas for the coefficients an and Products of Truncated Unitary Matrices bn of the recurrence pn+1(x)=(x − bn)pn(x) − anpn−1(x) for all the elements in the extended Hahn classes of discrete We study the joint singular value density of fixed or ran- orthogonal and q-orthogonal polynomial sequences. Such dom matrices multiplied with truncated unitary matrices coefficients are rational functions of n, in the first case, distributed by the induced Haar measure. We prove that and of qn, in the second case, which are determined by the the squared singular values of a fixed matrix multiplied initial terms b0,b1,b2,a1,a2. We consider the cases that with a truncated unitary matrix are distributed according yield simple rational functions. In addition to the well- to a polynomial ensemble, which is a special type of deter- known classical sequences we obtain many other examples, minantal point process. As a corollary, we apply this result 2 4 2 −1 such as bn = b0 and an = a1n or an =3n a1(4n −1) ,in to the situation where we multiply a truncated unitary ma- the discrete case. We also find the Bochner-type operator trix with a random matrix whose squared singular values for each example. [1] L. Verde-Star, Linear Alg. Appl. 440 are a polynomial ensemble and show that this structure (2014) 293–306. is preserved by the product. As an application we derive the joint singular value density of a product of truncated Luis Verde-Star unitary matrices and its kernel of the k-point correlation Universidad Autonoma Metropolitana function. The correlation kernel can be written as a double FA15 Abstracts 39

contour integral and using this integral representation we focusing on the case n =2. obtain hard edge scaling limits. Vincent Genest Dries Stivigny Centre de recherches math´ematiques KU Leuven Universit´edeMontr´eal [email protected] [email protected]

Sarah I. Post CP9 Centre de Recherches Math´ematiques Some Applications of Determninants to Orthogonal [email protected] Polynomials Luc Vinet We show how different type of structered matrices and their Centre de recherches mathematiques determiants contribute to the theory orthogonal polynomi- [email protected] als and related ordinary differential equations and provide new classes of orthogonal polynomials. Alexei Zhedanov Mikhail Tyaglov Donetsk Institute for Physics and Engineering, Ukraine Shanghai Jiao Tong University [email protected] [email protected] MS2 MS1 Repeated Integrals of the Coerror Function, Revis- ited Multivariate Orthogonal Polynomials and Inte- grable Systems Nonstandard Gaussian quadrature is applied to evaluate the repeated integral inerfc x of the coerror function for n ∈ In this talk we discuss multivariate orthogonal polynomials N0, x ∈ R in an appropriate domain of the (n, x)-plane. in the Euclidean space and multivariate orthogonal Lau- Relevant software in Matlab is provided, in particular two rent polynomials in the unit torus. Using the Gauss-Borel routines evaluating the function to an accuracy of twelve factorization of a moment matrix we consider three term resp. thirty decimal digits. relations, Christoffel-Darboux formulas, Darboux transfor- mations and the connection with a hierarchy of compatible Walter Gautshi nonlinear partial differential and difference equations of in- Purdue University tegrable type. [email protected] Manuel Manas Departamento de Fsica Te´orica II MS2 Universidad Complutense de Madrid Numerical Multivariate Polynomial Factorization manuel.manas@fis.ucm.es We present a method to extract the factors of a multivari- ate polynomial in floating point arithmetic. We establish MS1 the connection between the irreducible factors of a multi- Orthogonal Polynomials on the Unit Ball and variate polynomial and the Taylor expansion of its recipro- Fourth Order Partial Differential Equations cal. Based on this connection, a multivariate generalization of the qd-algorithm is employed to compute the irreducible In this work we study a family of mutually orthogonal poly- factors. Our iterative method does not require the input nomials on the unit ball with respect to an inner product polynomial to be square-free. Moreover, this approach pro- which includes an additional spherical term. First, we will vides additional insight to some problems in multilinear deduce connection formulas relating classical multivariate algebra. orthogonal polynomials on the ball and the new family of orthogonal polynomials. Then, using the representation of Wen-shin Lee, Annie Cuyt these polynomials in terms of spherical harmonics, alge- University of Antwerp braic, analytic and differential properties will be deduced. [email protected], [email protected]

Miguel Pinar MS2 University of Granada DLMF Standard Reference Tables on Demand [email protected] (DLMF Tables) (6)

Most current libraries and systems produce tables of func- MS1 tion values with limited information about accuracy. To Algebraic Interpretation of Multivariate q- address this void, NIST ACMD and the University of Krawtchouk Polynomials Antwerp CMA Group are collaborating to build the DLMF Standard Reference Tables web service to provide a stan- An algebraic interpretation of q-analogs of the multivari- dard of comparison for testing numerical software by com- ate Krawtchouk polynomials introduced by Tratnik will be puting, on demand, special functions to user-defined accu- presented. It will be shown that these polynomials in n racy with guaranteed error bounds. We will discuss the variables arise as matrix elements of unitary ”q-rotation” beta version recently released, plus current and proposed operators expressed as q-exponentials in slq(n + 1) gener- work. ators realized in terms of the ladder operators of n inde- pendent q-oscillators. The main ideas will be illustrated by Bonita V. Saunders 40 FA15 Abstracts

National Institute of Standards and Technology (NIST) Proc. Nat. Acad. Sci., 71(1974), 4082-4085). A study of [email protected] this final result in Bressoud’s paper reveals the Bressoud polynomials which in turn have various applications and Bruce Miller lead to open problems. Applied and Computational Mathematics Division National Institute of Standards and Technology George E. Andrews [email protected] Penn State University Department of Mathematics [email protected] Marjorie McClain Mathematical and Computational Sciences Division National Institute of Standards and Technology MS3 [email protected] Monodromy of Hypergeometric Functions in Sev- eral Variables Daniel Lozier National Institute of Standards and Technology The analytic continuation properties of single variable hy- [email protected] pergeometric values is well-known nowadays. The next challenge is to obtain similar results for the class of Andrew Dienstfrey A-hypergeometric functions. These were introduced by National Institute of Standards and Technology (NIST) Gel’fand, Kapranov and Zelevinsky in the late 1980’s and [email protected] contain the classical Appell and Lauricella functions as spe- cial cases. Annie Cuyt Frits Beukers University of Antwerp Utrecht University [email protected] [email protected] Stefan Becuwe, Franky Backeljauw University of Antwerp, Belgium MS3 [email protected], q-Bessel Functions and Rogers-Ramanujan Type [email protected] Identities

We show how q-Bessel functions of the first kind lead to MS2 Rogers-Ramanujan type identities. Carlotta considered a Near-Minimal Cubature Rules on the Disk special case of this and used it to give new proofs of the Rogers-Ramanujan identities. High-dimensional integration rules have received a lot of attention in a whole series of paper. The connection with Mourad Ismail multivariate orthogonal polynomials is usually present. We University of Central Florida present a unifying theory, including several of the existing [email protected] minimal rules, departing from the rather recent concept of spherical orthogonality. Use of the newly defined multivari- Ruimimg Zhang ate orthogonal functions leads to a structured set of Prony- Northwest A & F University structured equations from which the nodes and weights of [email protected] certain minimal cubature formulae can be computed.

Irem Yaman MS3 Gebze Technical University Mahler Measures of Hyperelliptic Families [email protected] I will overview some recent progress on the Mahler mea- Annie Cuyt sure of two-variable polynomials corresponding to special University of Antwerp elliptic families; these are known as Boyd’s conjectures. A [email protected] machinery, which was created in our joint work with Mat Rogers and further developed by Anton Mellit and Fran¸cois Brunault, allows one to relate the Mahler measures to the Brahim Benouahmane L-values of the underlying elliptic curves when the latter Universite Hassan II are parametrised by modular units. The talk will culminate Morocco with new results on Boyd’s conjectures for a hyperelliptic [email protected] family obtained in joint work with Marie Jos´eBertin(Paris 6).

MS3 Wadim Zudilin Bressoud Polynomials and the Rogers-Ramanujnan University of Newcastle Identities [email protected]

In 1983, D.M. Bressoud published An Easy Proof of the Rogers-Ramanujan Identities (J. Number Th., 16(1983), MS4 235-241). The elegance of his proof has been noticed by Equilibrium Measures and Their Support many. The conclusion of the paper is devoted to a general- ization of his result where he proves a polynomial identity Let Q(x) be an admissible external field. Let ν be the which in the limit yields two instances of the generalized signed equilibrium associated with Q(x). Suppose that the Rogers-Ramanjuan series/product identity (G.E. Andrews, positive part of ν consists of finitely many intervals, and ν FA15 Abstracts 41

has log-concave density there. We show that the support of moment-generating function and characteristic function. the equilibrium measure consists of finitely many intervals. Jorge C. Buescu Faculdade de Ciencias David Benko Universidade de Lisboa Univerity of South Alabama [email protected] Mobile, AL [email protected] Ant´onio Paixo ISEL [email protected] MS4 Asymptotically d-Energy Minimizing Sequences of Alexandra Symeonides Configurations on d-dimensional Conductors IST [email protected] We show that the sequence of configurations on a compact d set A ⊂ R with Ld(A) > 0andLd(∂A)=0(Ld is the Lebesgue measure), obtained by intersecting A with any MS5 d-dimensional lattice scaled by a factor going to zero is On Perturbations of the Toda Lattice asymptotically d-energy minimizing. When Ld(∂A) > 0, an asymptotically d-energy minimizing sequence is con- 1 We present the results of an analytical and numerical study structed. When A is contained in a d-dimensional C man- (with Nenciu) of the long-time behavior for certain Fermi- ifold, any asymptotically best-packing sequence of N-point Pasta-Ulam lattices viewed as perturbations of the com- configurations is asymptotically d-energy minimizing. pletely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Sergiy Borodachov Jacobi matrices. We also present our work (with Trogdon) Towson University where we solve the Cauchy problem for the Toda lattice by Baltimore, USA computing the Riemann-Hilbert formulation of the inverse [email protected] scattering transform.

Douglas Hardin, Edward Saff Deniz Bilman, Irina Nenciu Vanderbilt University University of Illinois at Chicago [email protected], ed- [email protected], [email protected] ward.b.saff@vanderbilt.edu Thomas Trogdon Courant Institute of Mathematical Sciences MS4 NYU Covering and Separation for Points on the Sphere [email protected]

The covering radius of an N-point set XN on the unit d+1 sphere in R is the radius of the largest spherical cap that MS5 contains no points from XN whereas the separation dis- Transition Asymptotics for the Painlev´eIITran- tance gives the least distance between two different points scendent in XN . In the theory of approximation and interpola- tion, the covering radius arises in the error of approxima- We consider real-valued solutions u = u(x|s),x ∈ R of 3 tion whereas good separation of points is generally associ- the second Painlev´eequationuxx = xu +2u which ated with the stability of an approximation or interpola- are parametrized in terms of the monodromy data s ≡ 3 tion method. We present resent results and open questions (s1,s2,s3) ⊂ C of the associated Flaschka-Newell system regarding the covering and separation properties of, in par- of rational ODEs. Our analysis describes the transition ticular, random points on the sphere. This talk is based asymptotics as x →−∞between the oscillatory power- on joint work with Josef Dick, Ed Saff, Ian Sloan, Yuguang like decay asymptotics for |s1| < 1 (Ablowitz-Segur) to the Wang and Rob Womersley. power-like growth behavior for |s1| = 1 (Hastings-McLeod) and from the latter to the singular oscillatory power-like Johann Brauchart growth for |s1| > 1 (Kapaev). It is shown that the transi- Technical University at Graz tion asymptotics is of Boutroux type, i.e. it is expressed in Graz, Austria terms of Jacobi elliptic functions. [email protected] Thomas J. Bothner Universit´edeMontr´eal MS4 [email protected] Characterization of Complex Variable Positive Def- inite Functions MS5 A holomorphic positive definite function f defined on a hor- On Non-Linearizable Boundary Value Problems for izontal strip of the complex plane is characterized as the the Defocusing Nonlinear Schr¨odinger Equation on Fourier-Laplace transform of a unique exponentially finite the Half-Line measure on R. This characterization unifies the integral representations in Bochner’s theorem on positive definite We study non-linearizable Dirichlet initial/boundary-value functions and Widder’s theorem on exponentially convex problems for the defocusing nonlinear Schr¨odinger equa- functions. Special interesting cases include the Gamma tion on the half-line x>0usingamodificationofthe function and the Riemann Zeta function. Further charac- unified transform method. We sidestep the global relation terizations are derived from the usual concepts of moment, by instead introducing an explicit approximation of the 42 FA15 Abstracts

Dirichlet-to-Neumann mapping for the problem suggested results in the field will be presented. by dispersionless theory. Accuracy is proven directly in the semiclassical/dispersionless limit, with the help of steepest Jeremy Wade descent methods. This is joint work with Zhenyun Qin, Pittsburg State University Fudan University. [email protected]

Peter D. Miller University of Michigan, Ann Arbor MS6 [email protected] A New Identity for Gegenbauer Polynomials and Reproducing Kernels for Multivariable Orthogonal Zhenyun Qin Polynomials Fudan university A new integral identity that expresses the Gegenbauer [email protected] λ μ polynomial Cn in terms of an integral of Cn ,withμ>λ, is established. It is used to derive concise formulas for re- MS5 producing kernels of orthogonal polynomials for two fam- ilies of weight functionsL: the product Gegenbauer weight Numerical Inverse Scattering for the BenaminOno functions on the unit cube and the generalized Gegenbauer Equation d weight functions on the unit ball of R . Numerical inverse scattering has proven a successful Yuan Xu tool for calculating the Korteweg–de Vries and nonlinear University of Oregon Schrdinger equations, which consists of calculating spec- [email protected] tral data for a differential equation (direct scattering) and solving a Riemann–Hilbert problem (inverse scatter- ing). The Benjamin–Ono equation does not fit immedi- MS6 ately in this form, as the direct scattering is a Riemann– Some Topics on Basis Function Approximation on Hilbert/differential equation and the inverse scattering is the Sphere a nonlocal Riemann–Hilbert problem involving an oscilla- tory integral term. In this talk we investigate numerics for In several cases, kernel-based methods for scattered data these more difficult problems. nicely frame within the context of approximation on Gelfand pairs. With the Euclidean motion group being Sheehan Olver the most prominent example, another geometrical setting The University of Sydney is given by zonal basis function approximation. The sphere [email protected] is tied to the pair (SOd,SOd−1) with Gegenbauer polyno- mials defining the spherical functions. Aiming at transfer- MS6 ring well-established results from radial basis function ap- proximation to the geometric setting of the sphere, certain Multivariate Meixner Polynomials and a Discrete properties of Gegenbauer polynomials are needed. While Model of the Two-dimensional Harmonic Oscillator some of these properties are already known, others lead to with su(2) Symmetry interesting problems dealing with this family of orthogo- A discrete model of the two-dimensional harmonic oscilla- nal polynomials. We will address these issues and explain tor based on the two-variable Meixner polynomials is pre- some of the results. The talk is based on joint work with sented. It is shown that the system is superintegrable and R. Beatson and Y. Xu. has su(2) symmetry. The interpretation of the d-variable Wolfgang zu Castell Meixner polynomials as matrix elements for SO(d, 1) rep- Helmholtz Zentrum Muenchen, German Research Center resentations on oscillator states is reviewed. for Envir Julien Gaboriaud Ingolstaedter Landstrasse 1, 85764 Neuherberg, Germany Universit´edeMontr´eal [email protected] [email protected] MS7 Vincent Genest Algorithms for An Interpolation Problem on the Centre de recherches math´ematiques Unit Circle Between Lagrange and Hermite Prob- Universit´edeMontr´eal lems [email protected] This piece of work is devoted to obtain formulae for an Jessica Lemieux interpolation problem on the Unit Circle (T). The interpo- University of Ottawa lation conditions gather the values of the polynomial at 2n [email protected] nodes and its first derivative at n nodal points; the nodal system is constituted by 2n equispaced points of modulus Luc Vinet 1. We give different solutions, between them we point out Centre de recherches mathematiques an explicit solution in terms of the natural basis and two [email protected] barycentric type formulae. All these formulae are suitable for numerical applications.

MS6 Elias Berriochoa, Alicia Cachafeiro Recent Developments in Hyperinterpolation University of Vigo [email protected], [email protected] Hyperinterpolation is an approximation technique devel- oped by Ian Sloan in 1995. In this talk, a survey of recent Jos´eGarc´ıA-Amor FA15 Abstracts 43

I. E. S. Valle Inclan, Xunta de Galicia gfl[email protected] [email protected]

MS8 MS7 Boundary Integrals and Approximations of Har- monic Functions Bernoulli Polynomials As a Base in Numerical Methods Steklov expansions for a harmonic function on a rectangle are analyzed. The value of a harmonic function at the cen- In theory, since the best approximation in L2[a, b] is unique, ter of a rectangle is shown to be well approximated by the all polynomials which are in L2[a, b] can give the same ap- mean value of the function on the boundary plus a very proximation of known functions. However, in application, small number (often 3 or fewer) of additional boundary it does not occur because we have computational error. We integrals. Similar approximations are found for the cen- know usually Legendre polynomials have the less compu- tral values of solutions of Robin and Neumann boundary tational error than other polynomials. We show Bernoulli value problems. These results are based on finding explicit polynomials create less computational error than Legendre expressions for the Steklov eigenvalues and eignfunctions. polynomials. For this reason, we introduced the Bernoulli This is joint work with Professor Giles Auchmuty. polynomials as trial functions. Giles Auchmuty University of Houston, USA Somayeh Mashayekhi [email protected] Mississippi state university [email protected] Manki Cho University of Houston [email protected] MS7 Approximation of Periodic Functions in Terms of Moduli of Continuity MS8 Universal Lower Bounds for Potential Energy of A 2p periodic function in Lp spaces is approximated by Spherical Codes the trigonometric polynomials of degree and the error of Minimal energy configurations (codes), maximal codes, approximation is given by norm minimozation of the norm and spherical designs have wide ranging applications in ||Tn(x) − f(x)||p. . The error defined above depends on various fields of science, such as crystallography, nanotech- the various properties of the function f. In this paper, we nology, material science, information theory, wireless com- obtain the error of approximation in terms of the modulus munications, etc. We derive universal lower bounds for the of continuity of the function f. We use some summability potential energy of spherical codes, that are optimal in the techniques to accelerate the rate of convergence. We also framework of the standard linear programming approach derive some corollaries from our results. and provide a characterization of when improvements are possible. Our bounds are universal in the sense of Cohn Uaday Singh and Kumar; i.e., they apply to any absolutely monotone Indian Institute of Technology Roorkee potential. Roorkee-247667 (India) [email protected] Peter Dragnev Indiana UniversityPurdue University Fort Wayne Soshal Saini [email protected] Indian Institute of Technology Roorkee Roorkee-247667 Peter Boyvalenkov [email protected] Institute for Mathematics and Informatics, Sofia, Bulgaria [email protected]

MS7 Douglas Hardin, Edward Saff A New Way to Compute Sine Integral Function Vanderbilt University [email protected], edward.b.saff@vanderbilt.edu Considering its wide application areas in science and engi- neering, efficient calculation of the sine integral function is essential. We present a new method to calculate the sine Maya Stoyanova integral function which is based on an accurate represen- Sofia University, Bulgaria tation of the sinc function as a sum of scaled cosines. The [email protected]fia.bg proposed method is as accurate as MATLAB’s sinint and is more efficient than both MATLAB’s sinint and trun- MS8 cated spherical Bessel function expansion, especially for large number of points. Asymptotics of Minimal Discrete Periodic Energy Problems Evren Yarman Let L be a d-dimensional lattice in Rd. For a parameter s> Schlumberger 0, we consider the asymptotics of N point configurations [email protected] minimizing the L-periodic Riesz s-energy as the number of points N goes to infinity. In particular, we focus on the case Garret Flagg 0

2 1+s/d 1+s/d C0N + C1N + o(N )asN →∞. The constants ence from the non vanishing background possess’s a fixed C0 and C1 depend only on s, d, and covolume of the lattice number of finite moments and derivatives. Using proper- L. ties of the scattering map of NLS we derive as a corollary an asymptotic stability result for initial data which are suf- Douglas Hardin ficiently close to the N-dark soliton solutions of NLS. Vanderbilt University [email protected] Robert Jenkins Dept. of Mathematics University of Arizona MS8 [email protected] Asymptotics for Maximal Polarization Configura- tions

For a compact manifold A in Euclidean space and a lower MS9 semi-continuous kernel K on A × A we consider the prob- Long-Time Asymptotics for a Discrete-Discrete In- lem of determining the asymptotic behavior of N-point N tegrable Equation configurations {xi}i=1 that maximize the minimum of the N potential i=1 K(x, xi)forx in A. We refer to such ex- The lattice potential KdV equation is a fully discrete inte- tremal problems as polarization problems (the terminology grable equation, whose initial value problem was recently Chebyshev problems is also used). We focus especially on s studied by Butler and Joshi. We obtain long-time asymp- Riesz s-kernels K(x, y)=1/|x − y| for s>0andtheir totics for this equation, using a suitable modification of connection with best covering problems on A. Results for the methods of Teschl’s group. The results show a strik- s>dim(A) are of particular interest. ing similarity with corresponding results for the continuous KdV equation, reinforcing the value of this discretization. Edward Saff Vanderbilt University David A. Smith edward.b.saff@vanderbilt.edu University of Crete [email protected] MS9 Saturated Bands in Many-Pole Riemann-Hilbert Problems MS9 Direct Scattering and Small Dispersion for the Many-pole Riemann-Hilbert problems arise in the analysis Benjamin-Ono Equation with Rational Initial Data of semiclassical limits for differential equations and orthog- onal polynomials. In current methods certain parameter We propose a construction procedure for the scattering ranges corresponding to saturated bands must be excluded data of the Benjamin-Ono equation with a rational initial for technical reasons. We present progress on obtaining condition, under mild restrictions. The construction pro- asymptotic results in these regions with applications to the cedure entails building the Jost solutions of the Lax pair nonlinear Schrodinger equation, the sine-Gordon equation, explicitly and use their analyticity properties to recover and discrete orthogonal polynomials. the reflection coefficient, eigenvalues, and phase constants. Robert J. Buckingham We finish by showing that this procedure validates certain Dept. of Mathematical Sciences well-known formal results obtained in the zero-dispersion The University of Cincinnati limit. [email protected] Alfredo N. Wetzel, Peter D. Miller Peter D. Miller University of Michigan, Ann Arbor University of Michigan, Ann Arbor [email protected], [email protected] [email protected] MS10 David A. Smith University of Crete An Historical Approach to Sobolev Orthogonal [email protected] Polynomials

Megan Stone We present a survey about analytic propeties of polyno- University of Arizona mials orthogonal with respect to a Sobolev inner prod- [email protected] uct. The distribution of their zeros, asymptotic proper- ties, spectral theory as well as some applications will be considered. Some open problems will be discussed. MS9 Long Time Asymptotics and Stability of Finite Francisco Marcellan Density Solutions of the Defocusing NLS Equation University of Carlos III of Madrid [email protected] We consider the Cauchy problem for the defocusing nonlin- ear Schr¨odinger equation (NLS). Using the ∂ generalization of the nonlinear steepest descent method of Deift and Zhou MS10 we derive the leading order approximation to the solution of NLS in the solitonic region of space–time |x| < 2t for Linearly Related Sequences of Continuous and Dis- large times and provide bounds for the error which decays crete Derivatives of Orthogonal Polynomial Se- as t ∞for a general class of initial data whose differ- quences. Connections with Sobolev Orthogonal FA15 Abstracts 45

Polynomial Sequences with respect to the Sobolev inner product In this talk we consider structure relations of the form  1  1 N M   m k f,g S := f(x)g(x)dμ0(x)+λ f (x)g (x)dμ1(x), ri,n D Qn−i+m(x)= si,nD Pn−i+k(x) , −1 −1 i=0 i=0 where λ is a non-negative constant and (dμ0,dμ1)isan where (Pn)n and (Qn)n are two sequences of orthogonal (M,N)-coherent pair of measures supported on the interval polynomials (OPs), continuous or discrete, and, for a non- − s [ 1, 1] with M and N fixed non-negative integer numbers. negative integer number s, D represents the continuous In this talk we discuss necessary and/or sufficient condi- 1,p derivative or a discrete difference-derivative operator of or- tions for the convergence in the W ([−1, 1], (dμ0,dμ1)) der s (the possibilities for D include the usual discrete op- norm of the Fourier expansion in terms of {qn,λ}n≥0,with erators Dω and Dq ). In the above relation, the orders of 1

MS10 The generating function Gd for the return probabilities in a (M,N)-Coherent Pairs of Linear Functionals and d-dimensional face-centered cubic lattice satisfies an ODE Jacobi Matrices whose order grows quadratically with d. Until recently only the ODEs for d ≤ 7 were known; using a recursive method We give a matrix interpretation of (M,N)-coherent pairs of for computing the coefficients of Gd, proposed by Zenine, linear functionals: An algebraic relation between the corre- Hassani, Maillard, we are able to go up to d = 11. These sponding Jacobi matrices associated with such functionals ODEs share many remarkable properties which we shall is stated, and the classical case is analyzed; The relation be- discuss in this talk. This is joint work with Jean-Marie tween the Jacobi matrices associated with (M,N)-coherent Maillard. pairs of order m and the Hessenberg matrix associated with the multiplication operator in terms of the basis of polyno- Christoph Koutschan mials orthogonal with respect to the Sobolev inner product RICAM Linz defined by such coherent pair is deduced. Austrian Academy of Sciences (AW) [email protected] Natalia C. Pinz´on-Cort´es Universidad Nacional de Colombia [email protected] MS11 Divisibility Properties of Sporadic Ap´ery-like Francisco Marcellan Numbers University of Carlos III of Madrid [email protected] It was shown by Gessel that the Ap´ery numbers, intro- duced by Ap´ery in his proof of the irrationality of ζ(3), are periodic modulo 8. We investigate this, and other divisibil- MS10 ity results, for Ap´ery-like numbers. For instance, we prove W (1,p) - Convergence of Fourier-Sobolev Expansions that the Almkvist–Zudilin numbers are periodic modulo 8 Associated to (M,N) Coherent Pairs of Measures as well. The ingredients of our proof are a multivariate ra- tional function whose diagonals are the Almkvist–Zudilin Let {qn,λ}n≥0 be the sequence of polynomials orthonormal numbers and a theorem originating with Furstenberg which 46 FA15 Abstracts

states that, modulo a fixed prime power, these values have 1437-1442, and in G.E Andrews, Partitions with early con- algebraic generating function and, hence, can be generated ditions, In Advances in Combinatorics Waterloo Workshop by a finite state automaton. As demonstrated recently by in Computer Algebra, W80 May. 26-29, 2011. Rowland and Yassawi, these automata can be computed mechanically. This talk includes joint work with Arian George E. Andrews Daneshvar, Pujan Dave, Amita Malik and Zhefan Wang Penn State University that was done as part of an Illinois Geometry Lab project. Department of Mathematics [email protected]

Armin Straub University of Illinois at Urbana-Champaign MS12 [email protected] Mock Theta Functions, Partial Theta Functions, and Their Ghosts - Rhoades

MS11 There is an association between mock theta functions and Positive Rational Functions and their Diagonals partial theta functions; they each carry half of the Fourier coefficients of a non-holomorphic modular form. In this An interesting phenomenon about some arithmetically sig- association some mysterious q-hypergeometric series may nificant sequences, like the sequence of Ap´ery numbers arise. We call these ”ghost terms”. This talk explains the (http://oeis.org/A005259), is that they are the diagonal association and gives hints of the curious structure in ghost sequences of the Taylor expansions of “interesting’ rational terms which remains without a theory. functions, and the positivity of the coefficients in the latter expansions are (conjecturally) determined by the positivity Robert Rhoades of the sequences. In my talk I will explain and illustrate Stanford University the phenomenon and also relate it to other classical results [email protected] in analysis, in particular, to some 60-year old theorems of H¨ormander. This is joint work with Armin Straub. MS12 Wadim Zudilin Mock Theta Functions and Quantum Modular University of Newcastle Forms [email protected] In this talk, I will describe several related recent results re- lated to mock theta functions. These functions have very MS12 recently been understood in a modern framework thanks Mock Modular Forms of Weight 5/2 and Partitions to the work of Zwegers and Bruinier-Funke. Here, we will revisit the original writings of Ramanujan and look at his We study the coefficients of a natural basis for the space original conception of these functions, which gives rise to of mock modular forms of weight 5/2 on the full modular a surprising picture connecting important objects such as group. The “shadow’ of the first element of this infinite generating functions in combinatorics and quantum mod- basis encodes the values of the partition function p(n). We ular forms. show that the coefficients of these forms are given by traces of singular invariants. These are values of modular func- Larry Rolen tions at CM points or their real quadratic analogues: cycle University of Cologne integrals of such functions along geodesics on the modu- [email protected] lar curve. The real quadratic case relates to recent work of Duke, Imamoglu, and Toth on cycle integrals of the j- function, while the imaginary quadratic case recovers the MS13 algebraic formula of Bruinier and Ono for the partition Painleve Equations - Nonlinear Special Functions function. The Painlev´e equations, discovered over a hundred years Nickolas Andersen ago, are special amongst nonlinear ordinary differential University of Illinois at Urbana-Champaign equations in that they are “integrable” due to their rep- [email protected] resentation as Riemann-Hilbert problems. The Painlev´e equations can be thought nonlinear analogues of the clas- sical special functions and have numerous remarkable prop- MS12 erties. In this talk I shall give an introduction to the Partition Identities and Mock Theta Functions Painlev´e equations and discuss some open problems.

This is joint work with Stephen Hill, a Penn State under- Peter Clarkson graduate. In 1961, Basil Gordon proved a sweeping gener- University of Kent alization of the Rogers-Ramanujan identities. His theorem Kent UK may be broadly characterized as identifying the generat- [email protected] ing function for partitions having specified difference con- ditions on the parts with the quotient of two theta func- tions. We shall provide a new class of partitions (similar MS13 to those studied by Gordon) where the generating function Numerics for Classical Applications of Riemann- is identified with the quotient of a Hecke-type theta series Hilbert Problems divided by the Dedekind eta function. The simplest case is related to one of the fifth order mock theta functions of We overview several classical problems that can be re- Ramanujan. The partitions in question are similar in kind duced to RiemannHilbert problems, falling into three cat- to those described in: G.E Andrews, Partitions with initial egories: integral representations, differential equations and repetitions, Acta Math. Sinica, English Series, 25(2009), inverse spectral problems. The integral representation of FA15 Abstracts 47

error functions and elliptic integrals can be rewritten in smallest. Since they are positive definite the study of the terms of scalar RiemannHilbert problems. The Stokes’ smallest eigenvalue is the study of the statistics near the phenomenon for Airy’s equation and monodromy prob- origin of the spectral axis. The simplest instance is the La- lems for Fuchsian differential equations lead to matrix Rie- guerre ensemble; in this case the fluctuations are described mannHilbert problems. The inverse spectral problem for by a determinant random point field defined in terms of Jacobi and Schrdinger operators can be solved via matrix the celebrated Bessel kernel. This behaviour is also proven RiemannHilbert problems depending on a parameter, the in the literature to be ?universal?. We consider a model of latter of which encodes the solution to the Kortewegde several coupled matrices of Laguerre type (with a specific Vries equation. In all three cases, applying numerics to interaction) and address the corresponding study of the the RiemannHilbert problem allows for efficient approxi- origin of the spectrum. We find a natural generalization of mation, that is uniformly accurate in the complex plane. the Bessel random point field to a multi-specie analog that Joint work with Thomas Trogdon involves special functions (Meijer-G). It is then natural to formulate a universality conjecture. Sheehan Olver The University of Sydney Marco Bertola [email protected] Concordia University, Canada [email protected]

MS13 Thomas Bothner Explorations of the Solution Space of the Fourth CRM, Concordia University Painlev Equation Canada [email protected] Solutions to the Painlev´eequations(PI -PVI)thatarefree of poles over extensive regions of the complex plane are of considerable interest, dating back to the tronqu´ee and MS14 tritronqu´ee solutions of PI (Boutroux 1913). This talk The Normal Matrix Model in the Supercritical presents parameter choices and initial conditions of PIV that exhibit these pole-free sectors, highlighting the exis- Regime, and Asymptotics of the Associated Or- tence of the families of solutions that are free from (or have thogonal Polynomials only a finite number of) poles on the real axis. The normal matrix model is a random matrix model with Jonah A. Reeger eigenvalues in the complex plane. The average character- USAF istic polynomials are orthogonal with respect to weighted jonah.reeger@afit.edu area measure on the full plane. However, the integrals that are involved in the orthogonality do not converge in many interesting cases, and then the orthogonality has to be re- Bengt Fornberg defined. We follow the approach of [Bleher and Kuijlaars, University of Colorado Adv. Math. (2012)] for the model with a cubic potential, Boulder where the orthogonality is redefined as orthogonality on [email protected] contours in the complex plane. The orthogonal polynomi- als admit a Riemann-Hilbert characterization of size 3x3, MS13 which was analyzed in the subcritical case by means of the Deift-Zhou method of steepest descent. Here we analyze Uniformly Accurate Computation of Painlev´eII the supercritical case, and we find the limiting behavior of Transcendents zeros. We also find a new critical behavior, and our results only hold up to this second criticality. The Riemann–Hilbert approach for the Painlev´e transcen- dents has proved to be a powerful and rigorous tool for Arno Kuijlaars asymptotic analysis. The approach makes use of the Deift– Katholieke Universitaet Leuven Zhou method of nonlinear steepest descent. Recently, this [email protected] approach has been adapted for numerical purposes and the resulting computations are seen to be uniformly accurate. In this talk, I will outline the method and describe the rig- Alexander Tovbis orous justification for the accuracy that is observed. This University of Central Florida is joint work with Sheehan Olver. Department of Mathematics [email protected] Thomas Trogdon Courant Institute of Mathematical Sciences NYU MS14 [email protected] Discrete Toeplitz Determinants and Their Appli- cations

MS14 We will discuss the asymptotics of the Toeplitz determi- The Statistical Behaviour of the Low-end Spectra nantswithdiscretemeasure.Wefirstshowhowonecan of Several Coupled Matrices and the Meijer-G ran- convert this problem to the asymptotics of continuous or- dom Point Field thogonal polynomials by using a simple identity. Then we apply this method to the width of non-intersecting pro- Universality results in the statistical behaviour of spectra cesses of several different types. of random matrices typically consider the fluctuations near a macroscopic point of the spectrum. In an ensemble of Zhipeng Liu positive definite matrices, there are two essentially differ- Courant Institute of Mathematical Sciences ent distinguished points; the largest eigenvalues and the NYU 48 FA15 Abstracts

[email protected] measure. A natural question to ask is what happens in the remaining case when α is a real number: namely when −α := k is a MS14 positive integer? In this case, it is not difficult to see that −k The Condition Number of the Critically–Scaled La- the Laguerre polynomials {Lm | m ≥ k} form a complete guerre Unitary Ensemble orthogonal set in the Hilbert space L2((0, ∞); x−ke−x). A not-so-obvious question to ask is whether there is a ‘natu- In recent computations with S. Olver, we demonstrated ral’ inner product in which the entire sequence of Laguerre universal fluctuations in the iteration count, called the halt- −k ∞ polynomials {Lm }m=0 are orthogonal? The answer, sur- ing time, of classical numerical algorithms with random prisingly, is yes. We will show that the Laguerre polynomi- initial data. In particular, we noticed this phenomenon in −k ∞ als {Lm }m=0 form a complete orthogonal set in a Sobolev the conjugate gradient algorithm for solving Ax = b where space Sk with an inner product of the form A>0. The random data is formed by letting A = XX∗  where X is an N × n matrix with iid entries distributed k ∞ (j) (j) according to some random variable D and n is scaled in a (f, g)Sk = f (x)g (x)dμj. critical manner. In this talk, I will present a limit theorem j=0 0 for the condition number of the Laguerre Unitary Ensem- ble (D is a complex Gaussian) with this critical scaling We will also discuss a self-adjoint operator Tk, generated by and relate it to the performance of the conjugate gradient the classical second-order Laguerre differential expression, { −k}∞ algorithm. This is joint work with P. Deift and G. Menon. in the space Sk having the Laguerre polynomials Lm m=0 as a complete set of eigenfunctions. Thomas Trogdon Courant Institute of Mathematical Sciences Lance L. Littlejohn NYU Department of Mathematics, Baylor University [email protected] lance [email protected]

MS15 MS15 Differential Equations for Discrete Sobolev Orthog- A Study of the Exceptional Xm-Jacobi Expression onal Polynomials for Extreme Parameter Choices The aim of this talk is to study differential properties of In 2009, G´omez-Ullate, Kamran, and Milson extended the orthogonal polynomials with respect to a discrete Sobolev well-established Bochner Classification (1929) by showing { }∞ bilinear form with mass points. We will focus on the La- that the only polynomial sequences pn n=1 which simul- guerre case, where the mass point is located at zero, and taneously form a complete set of eigenstates for a second- the Jacobi case, where the mass point can be located either order differential operator and are orthogonal with re- at -1 or 1, or at the two points at the same time. We con- spect to a positive Borel measure having positive moments struct the orthogonal polynomials using certain Casorati are the exceptional X1-Laguerre and X1-Jacobi polynomi- determinants. Using this construction, we prove that they als. We will focus on the exceptional X1-Jacobi polyno- are eigenfunctions of a differential operator. Moreover, the mials. The second-order differential expression has ratio- nal coefficients and as a result, there is no solution of de- order of this differential operator is explicitly computed in  ∞ (α,β) terms of the matrices which defines the discrete Sobolev gree zero. The X1-Jacobi polynomials Pˆn form n=1 bilinear form. a complete orthogonal set in the weighted Hilbert space 2 L ((−1, 1);w ˆα,β), wherew ˆα,β is a positive rational weight Manuel Dominguez de la Iglesia function. Restrictions are placed on parameters α and β;in Instituto de Matem´aticas, Universidad Nacional particular, α, β > 0. We are particularly interested in the Autonoma Mexi ‘extreme’ parameter choice when α = 0. In this situation,  ∞ [email protected] (α,β) 2 Pˆn are in the associated L space, but we may n=2  ∞ Antonio Duran (0,β) actually study the full sequence of solutions Pˆn Universidad de Sevilla n=0 [email protected] in a certain Sobolev space S. Jessica Stewart MS15 Department of Mathematics and Computer Science, Laguerre Polynomials and Sobolev Orthogonality Goucher Coll [email protected] When α>−1, the classical Laguerre polynomials α ∞ {Lm}m=0 form a complete orthogonal set in the Hilbert space L2((0, ∞); xαe−x). More generally, from classical MS15 properties of these polynomials, it is the case that, for each Orthogonal Polynomials for Learning non-negative integer n, these Laguerre polynomials form a Within the burgeoning field of machine learning kernel complete orthogonal set in a certain Sobolev space Wn with inner product based learning has produced not only state of the art al- gorithms but stands out for the depth of its mathematical  n ∞ sophistication and elegance. The promise of kernel learning (j) (j) (j) α+j −x (f, g)n = Sn f (x)g (x)x e dx (f,g ∈ Wn),for applications lies in the abundance of reproducing ker- j=0 0 nels. In practice however there is a dearth of computable kernels. Orthogonal polynomials provide a compelling way (j) where {Sn } are the Stirling numbers of the second kind. for producing useful learning kernels. Associated spectral When α ≤−1 but −α is not a positive integer, the La- theory gives key insight into the nature of the regulariza- guerre polynomials are orthogonal with respect to a signed tion properties for learning machines with Sobolev orthog- FA15 Abstracts 49

onal polynomial kernels. Well discuss general theory as case, it is easy to see that the positive part of a rational well as several specific examples used in commercial appli- series is again a rational series. While this is no longer true cations. in the multivariate case, it is still true there that the pos- itive part of a D-finite series is D-finite. This can be used Richard Wellman in combinatorics to show that certain generating functions Department of Mathematics, Westminster College are D-finite. In the talk, we will show how to compute the [email protected] positive part of a D-finite multivariate Laurent series using creative telescoping. This is joint work with Alin Bostan, Frederic Chyzak, Lucien Pech and Mark van Hoeij. MS16 Transformations of Hypergeometric Functions Manuel Kauers Johannes Kepler University It is well known that the generalized hypergeometric func- [email protected] tions pFq satisfy many useful functional identities includ- ing the Kummer, Pfaff or Euler transform. Less is known in higher dimensions. Some results of a similar kind are MS16 known for example for Appell’s functions and other func- tions. We are going to provide a generalization of Euler, About Some Identities by Bayad and Beck Involv- Kummer and “Analytic continuation’ transform for n di- ing Bernoulli-Barnes Numbers, Barnes Zeta Func- mensional hypergeometric function for arbitrary n. tions and Fourier Dedekind Sums

Petr Blaschke A. Bayad and M. Beck have recently derived new identities Mathematical Institut in Opava, Silesian university in involving the Bernoulli-Barnes numbers, the Barnes zeta Opava functions and the Fourier Dedekind sums. Using symbolic Czech Republic calculus methods, we will provide some extensions and in- [email protected] terpretations of these identities. We will also show a link between the Bernoulli-Barnes numbers and the p-Bernoulli numbers as recently introduced by Rahmani, which are re- MS16 lated to the generating function of some special zeta values. Computing the Probability of Collision for Short- term Space Encounters: a Symbolic-Numeric Ap- proach Christophe Vignat Dpt. of Physics, Universite d’Orsay The increasing number of space debris in Low Earth Or- Dpt. of Mathematics, Tulane University bits constitute a serious hazard for operational satellites. [email protected] In order to provide adequate collision avoidance strategies, it is important to determine the collision probability be- tween two orbiting objects. Three-dimensional Gaussian MS17 probability densities represent the position uncertainties of the objects. With some simplifying assumptions, the prob- A Generalized Lebesgue Identity in Ramanujan’s lem of computing the collision probability, for short-term Lost Notebook encounters between space-borne objects, is, in practice, re- duced to a two-dimensional integral of a Gaussian function The Lebesgue identity is one of the important identities over a bounded region in a plane normal to the relative in the theory of partitions and q-hypergeometric series (q- velocity vector (encounter frame). The method presented series). The identity has a free parameter. By dilations here is based on an analytical expression for the integral, of the base q, and choices of the free parameter, several derived by use of Laplace transform and properties of D- fundamental partition identities follow. In Ramanujan’s finite functions. The formula has the form of a product Lost Notebook there is an identity in two free parameters. between an exponential term and a convergent power se- By viewing this as an extension of the Lebesgue identity, ries with positive P-recursive coefficients. Analytic bounds we obtain new weighted partition theorems. Our approach on the truncation error are also derived. This allows for yields new combinatorial information about certain Hecke- an efficient and reliable numerical evaluation of the risk. Rogers type series, and also leads to new companions to This talk is based on R. Serra, D. Arzelier, M. Joldes, J.- Euler’s celebrated Pentagonal Numbers Theorem. B. Lasserre, A. Rondepierre and B. Salvy, A New Method to Compute the Probability of Collision for Short-term Space Krishnaswami Alladi Encounters, AIAA/AAS astrodynamics specialist confer- University of Florida ence, American Institute of Aeronautics and Astronautics, alladik@ufl.edu pages 1-7, 2014.

Mioara Joldes MS17 LAAS-CNRS 7 Avenue du Colonel Roche, 31077 Toulouse, Cedex 4 Exotic Bailey-Slater Spt-Functions and Hecke- France Rogers Double Series [email protected] We study SPT-crank type functions that arise from Bailey pairs and that have interesting arithmetic properties. We MS16 find representations of these functions in terms of infinite The Positive Part of Multivariate Infinite Series products or two-variable Hecke-Rogers double series. The method uses Bailey’s Lemma and conjugate Bailey pairs. The positive part of an infinite series is defined as the for- mal power series which is obtained from it by discarding all Frank Garvan, Chris Jennings-Shaffer the terms involving negative exponents. In the univariate University of Florida 50 FA15 Abstracts

fgarvan@ufl.edu, cjenningsshaffer@ufl.edu asymptotically and the zeros localized through the vanish- ing of a theta divisor on an appropriate hyperelliptic curve. Our approach starts from a new Hankel determinant rep- MS17 resentation for the square of the Vorob’ev-Yablonski poly- Rogers-Ramanujan Type Identities nomial. This identity is derived using the representation of Vorob’ev polynomials as Schur functions. The Rogers–Ramanujan identities and a number of closely related identities first appeared just over 120 years ago. Thomas Bothner Early contributors to the theory of Rogers–Ramanujan- CRM, Concordia University type identities included Rogers, Ramanujan, F. H. Jack- Canada son, W. N. Bailey, F. J. Dyson, and L. J. Slater. Since [email protected] their introduction, identities of Rogers–Ramanujan type have found applications in a variety of areas including the theory of partitions, statistical mechanics, and vertex oper- MS18 ator algebras. I will review some of the history of Rogers– Asymptotics of Large-Degree Rational Painleve-IV Ramanujan type identities, including some of the more re- Functions cent contributions which relied rather heavily on computer algebra. The Painlev´e-IV equation admits a family of rational so- lutions, indexed by two integers, that can be expressed in Andrew V. Sills terms of generalized Hermite polynomials. In the large- Georgia Southern University degree limit the zeros of these polynomials form remark- [email protected] able patterns in the complex plane resembling rectangles with arbitrary aspect ratios depending on how the indexing integers grow. Using Riemann-Hilbert analysis we analyt- MS17 ically determine the boundary of the zero/pole region for Partitions Associated with the Ramanujan/Watson these rational functions. Mock Theta Functions omega(q) and nu(q) Robert Buckingham Recently, George Andrews, Atul Dixit, and I have discov- University of Cincinnati ered very interesting partition theorems that are related to [email protected] the mock theta functions ω(q)andν(q). For instance, the generating function for partitions where each part is less than twice the smallest part equals qω(q). In this talk, I MS18 will present those discoveries and some related arithmetic Painleve I in the Cubic Random Matrix Model properties. In his talk we analyze the double scaling regime in the Ae Ja Yee asymptotics of the partition function in the cubic random Department of Mathematics matrix model. In particular, we show the appearance of Pennsylvania State University solutions of the Painlev I differential equation in this set- [email protected] ting. We also provide a uniform asymptotic expansion of the Painlev I function Ψ(ζ,λ,α) introduced by Kapaev George E. Andrews to describe the Riemann-Hilbert problem corresponding to Penn State University Painlev I. [email protected] Alfredo Dea˜no Atul Dixit Departamento de Matem´aticas Tulane University Universidad Carlos III de Madrid [email protected] [email protected]

Pavel Bleher MS18 Department of Mathematics, IUPUI Zeros of Large Degree Vorob’ev-Yablonski Polyno- [email protected] mials Via a Hankel Determinant Identity

It is well known that all rational solutions of the second MS18 Painlev´e equation and its associated hierarchy can be con- On Large Degree Rational Solutions of Painleve II structed with the help of Vorob’ev-Yablonski polynomi- als and generalizations thereof. The zero distribution of The inhomogeneous Painlev´e-II equation with parameter the aforementioned polynomials has been analyzed numer- m has a (unique) rational solution exactly when m is an ically by Clarkson and Mansfield and the authors observed integer. Motivated in part by the universal appearance a highly regular and symmetric pattern: for the Vorob’ev of these functions in a certain double-scaling limit involv- polynomials itself the roots form approximately equilat- ing the semiclassical sine-Gordon equation, we study their eral triangles whereas they take the shape of higher or- asymptotic behavior in the limit of large m by means of der polygons for the generalizations. Very recently Buck- steepest descent methods applied to a Riemann-Hilbert ingham and Miller completely analyzed the zero distribu- problem encoding them that naturally arises in the sine- tion of large degree Vorob’ev-Yablonski polynomials using Gordon theory. Joint work with Robert Buckingham a Riemann-Hilbert/nonlinear steepest descent approach to (Cincinnati). the Jimbo-Miwa Lax representation of PII equation. In our work, joint with Marco Bertola, we rephrase the same prob- Peter D. Miller lem in the context of orthogonal polynomials on a contour University of Michigan, Ann Arbor in the complex plane. The polynomials are then analyzed [email protected] FA15 Abstracts 51

Robert Buckingham Marco Bertola University of Cincinnati Concordia University, Canada [email protected] [email protected]

MS19 MS19 The Riemann-Hilbert Approach to Polynomials Asymptotics in the Complex Plane for Multiple La- Orthogonal with Respect to Complex Weight Func- guerre Polynomials tions There are two kinds of multiple Laguerre polynomials. The In this contribution we illustrate how the Riemann-Hilbert multiple Laguerre polynomials of the first kind have or- approach and the Deift-Zhou steepest descent method can thogonality conditions with respect to weights xαj e−x on be used to analyze the asymptotic behavior and zero dis- [0, ∞)for1≤ j ≤ r,whereαj > −1aresuchthat tribution of polynomials that are orthogonal with respect αi − αj ∈/ Z. Multiple Laguerre polynomials of the second to complex weight functions. Some examples studied re- kind have orthogonality properties with respect to xαe−cj x iωx cently include a Fourier-type weight function, w(x)=e on [0, ∞)for1≤ j ≤ r,wherecj > 0aresuchthatci = cj on [−1, 1], an exponential weight with a cubic potential on whenever i = j. Wielonsky and Lysov (Constr. Approx. suitable contours in the complex plane, and Bessel func- 28 (2008), 61–111) have obtained strong asymptotics for tions on the semiaxis [0, ∞). the multiple Laguerre polynomials of the second kind using the Riemann-Hilbert problem and the Deift-Zhou steep- Alfredo Dea˜no est descent method. We will obtain strong asymptotics Departamento de Matem´aticas of the multiple Laguerre polynomials of the first kind us- Universidad Carlos III de Madrid ing the same techniques. This is joint work with Thorsten [email protected] Neuschel.

Walter van Assche MS19 Katholieke Universitaet Leuven The Riemann-Hilbert Approach to Critical Phe- [email protected] nomena in the Two Matrix Model

In this talk, an overview will be presented on various results MS20 for the asymptotic analysis of the two matrix model, based A Particular Case of a Higher Order Sobolev-Type on the Riemann-Hilbert approach for the biorthogonal Inner Product of Orthogonal Polynomials in Sev- polynomials that integrate this model. Particular emphasis eral Variables will be on recent results on the classification of the critical phenomena that can occur in the even quadratic/quartic We consider sequences of polynomials of several variables, case and the even quartic/quartic case. orthogonal with respect to the Sobolev-type inner product given in, which is obtained by adding to an inner stan- Maurice Duits dard product, the gradient operator of order j evaluated Mathematics Department in a particular point. We present an expression for the Stockholm University perturbed orthogonal polynomials in terms of the original [email protected] polynomials and a particular example on the unit ball in which we analize the asymptotic behaviour of the Kernel associated to the Sobolev-type polynomials. MS19 Asymptotics of Orthogonal Polynomials with Com- Herbert Due˜nas plex Varying Weight: Critical Point Behaviour and Departamento Matematicas, Universidad Nacional the Painleve Equations Colombia [email protected] We study the asymptotics of recurrence coefficients for complex monic orthogonal polynomials πn(z)withthe Wilmer M. G´omez quartic exponential weight Universidad Pedag´ogica y Tecnol´ogica de Colombia   z2 tz4 [email protected] exp −n + , 2 4 where t ∈ C and N →∞. We consider neighborhoods of MS20 − 1 1 1 the critical points t0 = 12 , t1 = 15 and t2 = 4 ,wherethe Sobolev Orthogonal Polynomials on the Unit Ball subleading terms can be expressed via Painlev´e transcen- via Outward Normal Derivatives dents. These subleading terms can become dominant near the poles of the corresponding Painlev´e transcendents. We The purpose of this work is to analyze a family of mutu- use the nonlinear steepest descent analysis for Riemann- ally orthogonal polynomials on the unit ball with respect Hilbert Problems to describe the recurrence coefficients in to an inner product which involves the outward normal full neighborhoods of tj , j =0, 1, 2, including the location derivatives on the sphere. Using the representation of these of the poles. We also provide the global (in the t-plane) polynomials in terms of spherical harmonics, algebraic and “phase diagrams, where the recurrence coefficients exhibit analytic properties will be deduced. First, we will get con- different asymptotic behaviors (Stokes’ phenomenon). nection formulas relating classical multivariate orthogonal polynomials on the ball with our family of Sobolev orthog- Alex Tobvis onal polynomials. Then explicit expressions for the norms Department of Mathematics will be obtained, among other properties. University of Central Florida [email protected] Lidia Fernandez 52 FA15 Abstracts

Departamento Matematica Aplicada, Universidad where Φn(z) are the monic orthogonal polynomials with Granada respect to the inner product f,g b,t =(1− t) f,g b + [email protected] tf(1) g(1).

Antonia Delgado Alagacone S. Ranga Universidad de Granada Departamento de Matematica Aplicada [email protected] Computacao,IBILCE-UNESP [email protected] Teresa E. P´erez Universidad de Granada MS21 Departamento de Matem´atica Aplicada [email protected] A Human Proof of Gessel’s Lattice Path Conjec- ture Miguel Pinar Gessel walks are lattice paths confined to the quarter plane University of Granada that start at the origin and consist of unit steps going ei- [email protected] ther West, East, South-West or North-East. In 2001, Ira Gessel conjectured a nice closed-form expression for the number of Gessel walks ending at the origin. In 2008, MS20 Kauers, Koutschan and Zeilberger gave a computer-aided On Asymptotic Properties of Sobolev Orthogonal proof of this conjecture. The same year, Bostan and Kauers Polynomials on the Unit Circle showed, again using computer algebra tools, that the com- plete generating function of Gessel walks is algebraic. In In this contribution, we study the sequence of polynomials this talk, I will report on the first human proofs of these orthogonal with respect to the Sobolev inner product two results, obtained in collaboration with Irina Kurkova  and Kilian Raschel. The proofs are derived from a new (j) (j) f,g S := f(z)g(z)dμ(z)+λf (α)g (α), expression for the generating function of Gessel walks in T terms of Weierstrass zeta-functions. where μ is a nontrivial probability measure supported on Alin Bostan the unit circle, α is a complex number, λ is a positive real INRIA Saclay Ile-de-France number, and j is a positive integer. In particular, we an- [email protected] alyze some asymptotic properties of such polynomials and the behavior of their zeros when n and λ tend to infinity, respectively. We also provide some numerical examples to MS21 illustrate the behavior of these zeros with respect to α. Numerical Evaluation of Contour Integrals for Computation of Stirling Numbers Luis E. Garza Departamento Matematicas,Universidad Nacional Flajolet and Prodinger (SIAM J. Discrete Math, 12:155– Colombia 159, 1999) proposed an interpolation of Stirling partition [email protected] numbers by the integral definition  − Francisco Marcellan x (x 1)! 1 − y−1 dz = exp(z)(exp(z) 1) x . University of Carlos III of Madrid y (y − 1)! 2iπ H z [email protected] Because the Γ functions are singular at negative integers this is not entirely satisfactory and some authors have pro- Kenier Castillo posed instead a related integral. This talk explores these University Carlos III de Madrid issues and discusses an explicit evaluation by numerical [email protected] means, using a parametrization of the Hankel contour that is related to the Tree T function (a cognate of Lambert W more suited to combinatorial applications, as is well MS20 known). Orthogonal Polynomials with Respect to a Sobolev Inner Product on the Unit Circle Rob Corless University of Western Ontario in London We look at some information concerning the orthogonal [email protected] polynomials with respect to the Sobolev inner product

  David Jeffrey f,g S =(1− t) f,g b + tf(1) g(1) + κ f ,g b+1, Department of Applied Mathematics where 0 ≤ t<1, κ ≥ 0, Re(b) > −1/2and University of Western Ontario djeff[email protected]  2π τ(b) iθ iθ π−θ Im(b) 2 Re(b) f, g b = f(e ) g(e )(e ) (sin (θ/2)) dθ.Yang Wang 2π 0 University of Western Ontario b+¯b 2 [email protected] 2 |Γ(b+1)| Here, τ(b)= Γ(b+¯b+1) is such that 1, 1 b =1and hence also 1, 1 S = 1. For example, the monic orthogonal polynomials Sn with respect to the inner product f,g S MS21 satisfy Utility Maximization and Symbolic Computation

Sn(z)+anSn−1(z)=Φn(z),n≥ 1, We give a brief introduction to utility maximization, which FA15 Abstracts 53

is a central topic of mathematical finance. Recently, there University of Illinois at Urbana-Champaign has been a lot of work on extending the classical frame- [email protected], [email protected], za- work by modelling the bid-ask spread. For small spread, [email protected] asymptotic expansions of the economically relevant quan- tities can be obtained. The special functions in our talk are the value functions of the optimization problems. They MS22 are characterized by Riccati ODEs with free boundary, and On Theta Quotients Generating Graded Algebras have explicit expressions in some cases. Computer algebra, of Modular Forms in particular symbolic manipulation of polynomials, is very useful to obtain asymptotic expansions of optimal strate- We construct theta quotients that generate graded alge- gies, optimal utility, and trading volume. bras of modular forms on principal congruence subgroups of prime level. A variety of consequences ensue, includ- Stefan Gerhold ing coupled systems of differential equations for sums of Financial and Actuarial Mathematics twisted Eisenstein series analogous to those on the full Vienna University of Technology modular group. [email protected] Tim Huber Department of Mathematics MS21 University of Texas-Pan American On 2F1-type Solutions of Second Order Linear Dif- [email protected] ferential Equations

Differential equations with 2F1-type solutions are very MS22 common in Mathematics and they also occur in several Cubic Modular Equations in Two Variables areas such as Physics and Combinatorics. Given a second order linear differential operator L ∈ C(x)[∂], we want to By adding certain equianharmonic elliptic sigma func- find a 2F1-type solution of the form tions to the coefficients of the Borwein cubic theta func-  tions, an interesting set of six two-variable theta func- exp( rdx) · 2F1(a, b; c; f) tions may be derived. These theta functions invert the 1 1 1 | F1 3 ; 3 ; 3 ;1x, y case of Appell’s hypergeometric function ∈ ∈ and satisfy several identities akin to those satisfied by the where r, f Q(x), and a, b, c Q. Thisformisboth Borwein cubic theta functions. Previous work on these more and less general than in prior work. In prior work, functions is extended and put into the context of Ramanu- solutions involving a sum of two 2F1’s were also considered, jan’s modular equations, resulting in a simpler derivations however, f was restricted to rational functions, while our as well as several new modular equations for Picard mod- method allows algebraic functions. ular forms.

Erdal Imamoglu Dan Schultz Department of Mathematics Pennsylvania State University Florida State University [email protected] [email protected]

Mark van Hoeij MS22 Florida State University Special Values of Trigonometric Dirichlet Series Tallahassee, FL [email protected] In his first letter to Hardy and in several entries of his note- books, Ramanujan recorded many evaluations of trigono- metric Dirichlet series such as MS22 ∞ r 2m B2(r−m) Ramanujan, Voronoi Summation Formula, Circle coth(πn) 1 2r−1 − m+1 B 2r−1 = (2π) ( 1) − , and Divisor Problems and Some Modular Trans- n=1 n 2 m=0 (2m)! (2(r m))! formations which, in fact, goes back to Cauchy and Lerch. A recent ex- On page 336 in his Lost Notebook, Srinivasa Ramanujan ∞ sec(πnτ) ample is the secant Dirichlet series ψs(τ)= n=1 ns , proposed an identity that may have been devised to attack for which Lal´ın, Rodrigue and Rogers conjecture, and par- a divisor problem. Unfortunately, the identity is vitiated √ tially prove, that its values ψ2m( r), with r>0 rational, by a divergent series appearing in it. We present here a cor- are rational multiples of π2m. We give an overview of spe- rected version of Ramanujan’s identity. This study has a cial values of such Dirichlet series and their connection with natural connection with the Voronoi summation formula. the theory of modular forms. This talk includes joint work We also obtain a one-variable generalization of two dou- with Bruce C. Berndt. ble Bessel series identities of Ramanujan on page 335 of the Lost Notebook which are intimately connected with Armin Straub the circle and divisor problems. Finally, we also obtain a University of Illinois at Urbana-Champaign new modular-type transformation involving infinite series [email protected] of Lommel functions. Such a transformation is extremely rare, and is the only known example of its kind. MS23 Atul Dixit Painleve Equations and Orthogonal Polynomials Tulane University [email protected] In this talk I shall discuss semi-classical orthogonal poly- nomials arising from perturbations of classical weights. It Bruce Berndt, Arindam Roy, Alexandru Zaharescu is shown that the coefficients of the three-term recurrence 54 FA15 Abstracts

relation satisfied by the polynomials can be expressed in components for |t| >tcr. Moreover, the strong asymp- terms of Wronskians which involve special functions. These totics of orthogonal polynomials with respect to measures 2d d d Wronskians are related to special function solutions of the of the form e−|z| +tz +tz dA(z) are obtained, where t is Painlev´e equations. Using this relationship recurrence rela- complex and dA is the area measure on the plane. Since tion coefficients can be explicitly written in terms of exact the orthogonality can be written in terms of contour in- solutions of Painlev´eequations. tegrals, the asymptotics are found via a Riemann–Hilbert problem. In particular, the Cauchy transform of the nor- Peter Clarkson malized counting measure of the zeroes converges to that University of Kent of the equilibrium measure in the exterior of the support Kent UK (based on joint works with T. Grava and D. Merzi). [email protected] Ferenc Balogh SISSA Trieste MS23 [email protected] Determinantal Representations of Exceptional Or- thogonal Polynomials MS24 Exceptional orthogonal polynomials represent an escape to Asymptotics for the Partition Function in Two-cut the Bochner Classification Theorem by not allowing poly- Random Matrix Models nomials of certain degrees as eigenfunctions to the corre- sponding differential expressions. Much of their fascinat- I will present a new method to obtain asymptotics for the ing rich structure had been exposed over the past 6 (or partition function in two-cut random matrix models, based so) years, and still more interesting research is being con- on Riemann-Hilbert problems. The method enables us ducted. We focus on the exceptional X1−Laguerre poly- to re-derive potential-dependent terms which were known nomials, which do not contain the constant polynomial. In in the physics literature, but also to evaluate rigorously particular, we find determinantal representations in terms potential-independent terms. The talk will be based on of moments where the first row of the determinant is ad- joint work with T. Grava and K. McLaughlin. justed to implement the ’exceptional condition’. A slight alteration of the moments greatly simplifies the represen- Tom Claeys tation. Department of Mathematics [email protected] Constanze Liaw Baylor University Tom Claeys USA Universit´e catholique de Louvain constanze [email protected] [email protected]

MS23 MS24 On the Alternative Discrete Painleve I Recent Developments in the Large-N Analysis of Correlation Functions in the Quantum Separation During this talk I will discuss special solutions of the al- of Variables Method ternative discrete Painlev´e-I equation. At the centre will be the uniqueness of the positive solution, which some- The scalar products and certain correlation functions of how links to the recurrence coefficients of exponential cubic models solvable by the quantum separation of variables can weights. beexpressedintermsofN-fold multiple integrals which can be thought of as the partition function of a one dimen- Ana Loureiro sional gas of particles evolving on a curve C,trappedin University of Kent an external potential V and interacting through repulsive [email protected]  two-body interactions of the type ln sinh[πω1(λ − μ)] ·  Peter Clarkson sinh[πω2(λ − μ)] . The choice of the curve C and of the University of Kent Kent UK confining potential V determines a given model. The anal- [email protected] ysis of the large-N asymptotic behaviour of these integrals is of interest to the description of the continuum limit of the integrable model. In this talk, I shall report on recent Walter van Assche developments in the large-N analysis of such integrals and Katholieke Universitaet Leuven discuss, on some specific examples, the form taken by the [email protected] asymptotics. Part of the results that I will present issue from a joint work with G. Borot and A. Guionnet.

MS24 KarolK.Kozlowski Orthogonal Polynomials for a Class of Measures Institut de Math´ematiques de Bourgogne with Discrete Rotational Symmetries in the Com- [email protected] plex Plane

Normal matrix models for external potentials of the form MS25 2n d d |z| + tz + tz are considered with integers 0 ≤ d ≤ 2n. Deformed Semi-classical Discrete Orthogonal Poly- A symmetry reduction procedure is used to find the equi- nomials librium measure for all values of n, d and t.Forfixedn and d, there is a critical value |t| = tcr such that the sup- In this talk a study families of discrete orthogonal poly- port is simply connected for |t|

linear type difference equation with polynomial coefficients Orthogonal Laguerre Polynomials is presented. The discrete dynamical systems, obtained as a result of deformations of the recurrence relation coeffi- We discuss interlacing properties of zeros of polynomials cients of the orthogonal polynomials related to the above of consecutive degree in sequences of Laguerre polynomials referred Stieltjes functions is derived. (α) ∞ {Ln }n=0 characterized by −2 <α<−1. Stieltjes inter- Am´ılcar Branquinho lacing between the zeros of qm and pn+1,m≤ n, where ∞ UNIVERSIDADE DE COIMBRA {qn}n=0 is a sequence of quasi-othogonal Laguerre polyno- ∞ [email protected] mials and {pn}n=0 is an orthogonal Laguerre sequence, is also investigated. Upper and lower bounds for the negative (α) zero of Ln , −2 <α<−1, are derived. MS25 Integral Transforms of d-orthogonal Polynomial Se- quences

This talk is mainly focused on index type integral trans- forms that map certain d-orthogonal polynomial sequences into another d-orthogonal polynomial sequences. It turns out that in several cases the weights are semiclassical.

Ana Loureiro University of Kent [email protected]

MS25 The Correspondence between the Askey Table of Orthogonal Polynomial Systems and the Sakai Scheme of Discrete Painlev´eEquations

The Askey Table is a classification of the hypergeomet- ric and basic hypergeometric orthogonal polynomial sys- tems in a single variable, whose members possess a num- ber of characteristic and defining properties. On the other hand the Sakai Scheme is a classification of the non- linear, integrable Painlev´e equations and their difference, q-difference and elliptic analogs, based upon algebraic- geometric ideas. It is possible to bring these two into cor- respondence whereby the Askey Table represents a base, trivial level and the Sakai Scheme is its first deformation, the first of the multi-variable extensions. In the process of doing so we solve a number of related problems: an algo- rithmic construction of Lax pairs for the discrete Painlev´e equations, the construction of an explicit sequence of clas- sical solutions to the discrete Painlev´e equations with use- ful applications to probabilistic models such as found in random matrix theory or random tiling models. In addi- tion the correspondence provides insight into a geometrical reformulation of the Askey Table itself akin to the root sys- tem symmetries of the hypergeometric functions.

Nicholas Witte University of Melbourne [email protected]

MS25 ”Abstract” Classical Polynomials: Open Problems

We discuss open problems arising in classification of ”ab- stract” classical orthogonal polynomials. These polynomi- als satisfy some general ”umbral” eigenvalue problems with unknown symbols of operators.

Alexei Zhedanov Donetsk Institute for Physics and Engineering, Ukraine [email protected]

MS26 Interlacing and Bounds for Zeros of Quasi- 56 FA15 Abstracts

Kathy A. Driver University of Cape Town Kathy A. Driver South Africa University of Cape Town [email protected] South Africa [email protected] MS27 Martin E. Muldoon The Quantum Superalgebra ospq(1|2) and a q- Department of Mathematics & Statistics generalization of the Bannai-Ito Polynomials York University [email protected] The Racah problem for the quantum superalgebra ospq(1|2) is considered. A quantum deformation of the Bannai-Ito algebra is realized by the intermediate Casimir MS26 operators entering the Racah problem. A q-generalization Weighted Norm Inequalities for Some Special of the Bannai-Ito polynomials is presented. The relation Functions between these basic polynomials and the q-Racah/Askey- Wilson polynomials is discussed. A method of obtaining new weighted norm inequalities for generalized hypergeometric functions and special functions Vincent Genest of hypergeometric type will be presented. It is based on Centre de recherches math´ematiques the limit version of the sequence of weighted convolution Universit´edeMontr´eal inequalities generated by a seminorm problem for formal [email protected] power series and its binomial solution. It will be shown that the limit inequality involves some deep properties of Luc Vinet the Eulerian integrals and Bernstein polynomial relation. Centre de recherches mathematiques Several related examples and applications of our method [email protected] will be discussed. Alexei Zhedanov Arcadii Grinshpan Donetsk Institute for Physics and Engineering, Ukraine University of South Florida [email protected] Tampa, FL, USA [email protected] MS27 Separation of Variables, Superintegrability and MS26 Bˆocher Contractions Meijer’s G Function and Fox’s H Function Near a Regular Singularity Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd In the talk, we will discuss some new properties of Mei- p,0 p,0 order superintegrable systems in 2 dimensions as special jer’s G function Gp,p and Fox’s H-function Hq,p . The first cases. Distinct superintegrable systems and their quadratic function has been studied by various authors under dif- algebras are related by geometric contractions, induced ferent names and our first goal is to reveal the connections by generalized In¨on¨u-Wigner Lie algebra contractions and between those investigations. In particular, we will present these contractions have important physical and geometric new formulas for the expansion coefficients in the vicinity consequences, such as the Askey scheme for hypergeomet- of the singular point at 1. The Fox H-function does not ric orthogonal polynomials. This approach can be unified satisfy any known differential equation but it has a similar by ideas first introduced in the 1894 thesis of Bˆocher to type of singularity at a finite real point (which need not be study separable solutions of the wave equation. unity). We will discuss its behavior in the neighborhood of this point. Further, we present new integral and functional Willard Miller Jr equation for both functions and inequalities for some other University of Minnesota cases of Meijer’s G-function. The talk will reflect the joint School of Mathematics work with Elena Prilepkina. [email protected]

Dmitry Karp Russian Academy of Sciences MS27 Vladivostok, Russia Exceptional Orthogonal Polynomials, Wronskians, [email protected] and the Darboux Transformation

Exceptional orthogonal polynomials (so named because MS26 they span a non-standard polynomial flag) are defined Interlacing of Zeros of General Laguerre Polynomi- as polynomial eigenfunctions of Sturm-Liouville problems. als By allowing for the possibility that the resulting sequence of polynomial degrees admits a number of gaps, we ex- We consider interlacing properties of the real zeros of the tend the classical families of Hermite, Laguerre and Jacobi. (α) Laguerre polynomial Ln (x), in the case α<−1. The In recent years the role of the Darboux (or the factoriza- main tool used is the Sturm comparison theorem. tion) transformation has been recognized as essential in the theory of orthogonal polynomials spanning a non-standard Martin E. Muldoon flag. In this talk we will focus on exceptional Hermite poly- Department of Mathematics & Statistics nomials: their regularity properties, asymptotics of zeros York University and their relation to the recent conjecture that ALL excep- [email protected] tional orthogonal polynomials are related via factorization FA15 Abstracts 57

transformations to classical orthogonal polynomials. [email protected]

Robert Milson Dept. Mathematics & Statistics MS28 Dalhousie University [email protected] Building DLMF I will describe the technical processes and issues involved in David Gomez-Ullate developing and building the Digital Library of Mathemat- Complutense University ical Functions (DLMF). These include: use of LAT Xfor [email protected] E authoring for print; LATEXML to convert to web formats with Presentation MathML; semantic markup for math- MS27 aware search and with an eye to future Content MathML. The applicability of these techniques to Digital Mathemat- Doubling-up Hahn Polynomials: Classification and ical Libraries and the web publishing of mathematics will Applications be addressed, as well as future work.

Joint work with J. Van der Jeugt. We examine how two sets Bruce Miller ˆ ˆ of Hahn polynomials Qn(x; α, β, N)andQn(x;ˆα, β,N)can Applied and Computational Mathematics Division be combined into a single set of (discrete) orthogonal poly- National Institute of Standards and Technology nomials Pn(x)(n =0, 1,...,N+ Nˆ + 1). Our analysis uses [email protected] (new and old) shift operator relations for Hahn polynomi- als, coming from contiguous relations. This investigation gives rise to new classes of (rather pretty) tridiagonal ma- MS28 trices with a closed form spectrum, and has applications in finite quantum oscillators or in linear spin chains. The Mathematica Computable Library of Special Function Identities Roy Oste,JorisVanDerJeugt Ghent University We report on the enhancement and extension of the sub- [email protected], [email protected] stantial body of special function identities originally col- lected on the Wolfram Functions Site. A greatly aug- mented set of identities including a number of new func- MS28 tions and mathematical constants has now been integrated into Wolfram|Alpha, making finding and working with Overview of Digital Mathematics Libraries (DML) these identities easier than ever before. In addition, using and the NIST Digital Repository of Mathematical the computable data framework pioneered for Mathemat- Formulae (DRMF) ica 10, this collection will soon be available in Mathematica The DRMF is designed for a mathematically literate au- itself. dience and should (1) facilitate interaction among mathe- maticians and scientists interested in compendia OPSF for- Eric Weisstein mulae data; (2) be expandable, allowing for input of new Wolfram Research, Inc. formulae from the literature; (3) use context-free semantic [email protected] markup; (4) have a user friendly, consistent, and hyper- linkable viewpoint and authoring perspective; (5) perform math-aware search; and (6) use MathML for easily read, MS29 scalably rendered, content driven mathematics. A DML Resurgence and Special Functions overview will be given. The talk aims at presenting advances on some special func- Howard Cohl tions from the viewpoint of resurgence theory. Among Applied and Computational Mathematics Division other examples, the Painlev´e transcendents will be par- National Institute of Standards and Technology ticularly adressed. [email protected] Eric Delabaere Universit´edAngers MS28 [email protected] Steps Toward Realizing a World Information Sys- tem for Digitally Organized Mathematics MS29 The research mathematics available online has greatly in- creased over the last decade and a half; the methods Accelerating the Computation of Special Functions for describing, linking and discovering these resources has Using Hybrid Hyper-Borel-Pad´e Approximations evolved much more slowly. A Global Digital Mathemat- ical Library Working Group was formed at the 2014 In- In this talk we will unify the two current techniques of hy- ternational Congress of Mathematicians. Concrete ideas perasymptotic expansions and Borel-Pad´e summation to for creating a community-owned mathematical information accelerate the high accuracy computation of special func- system for the digital age of science better supporting ad- tions and their associated Stokes constants. Examples will vanced research in mathematics, and steps already taken be given demonstrating the advantages, and disadvantages, will be discussed. of the techniques.

Patrick Ion Chris Howls American Mathematical Society Southampton University 58 FA15 Abstracts

[email protected] recurrence relation from the determinacy or indeterminacy of the original OPS. We will also give new results of this sort. MS29 The Resurgence Properties of the Bessel and Han- Theodore Chihara kel Functions of Large Order and Argument Purdue University [email protected] We reconsider the classical large order and argument asymptotic series of the Bessel and Hankel functions due to Debye. Employing the reformulation of the method MS30 of steepest descents by Berry and Howls, we derive From Indeterminate to Determinate resurgence-type formulas for the error terms of these ex- pansions. Using these new representations of the remain- In the talk, I’ll discuss various solutions to indeterminate der terms, we obtain numerically computable bounds and moment problems and explain how much (or little) one exponentially improved versions of Debye’s series, together needs to modify the measure in order to change the nature with asymptotic expansions for their late coefficients. Our of the problem. I’ll also discuss a number of conditions analysis also provides a rigorous treatment of the formal that usually won’t result in determinacy. This is mainly results derived earlier by Dingle. done through examples and counterexamples.

Gergo Nemes Jacob S. Christiansen Central European University University of Lund [email protected] [email protected]

MS29 MS30 Exponentially Small Difference Between the Eigen- Spectral Properties of Unbounded Jacobi Matrices values am and bm+1 of Mathieu’s Equation and Chihara’s Problem

d2w(z) We consider Mathieu’s equation 2 + Let μ be a determinate measure on the real line. Assume  dz ∞ 2 { } λ − 2h cos 2z w(z) = 0, which has special solutions the orthonormal polynomials pn n=0 with respect to μ called Mathieu functions of integral order when λ = am or satisfy ≡ ≡ λ = bm+1,form a positive integer. When h →∞,these p−1(λ) 0,p0(λ) 1, special eigenvalues have the same asymptotic expansions λpn(λ)=an−1pn−1(λ)+bnpn(λ)+anpn+1(λ)(n ≥ 0) to all orders in descending powers of h. I will present rigorous asymptotics to obtain the exponentially small for sequences {an} and {bn}. The Chihara problem is the difference between am and bm+1 as h →∞. following: assume

2 Karen Ogilvie an 1 lim bn = ∞, lim = The University of Edinburgh n→∞ n→∞ bnbn+1 4 [email protected] and the smallest accumulation point ρ of supp(μ)isfi- nite. Find additional conditions which imply supp(μ) ⊇ MS30 [ρ, +∞). We are going to present a solution to this prob- Recurrence Relations and Hamburger and Stieltjes lem. Moment Problems Grzegorz Swiderski Consider an orthogonal polynomial sequence (OPS) Instytut Matematyczny, Uniwersytet Wroclawski { }∞ Pn(x) n=0 which satisfies the three term recurrence re- [email protected] lation − − Pn(x)=(x cn)Pn−1(x) λnPn−2(x),cn real,λn+1MS31> 0,n=1, 2,..., P0(x)=1,P−1(x)=0. Computer Algebra and Special Functions Inequal- ities We are concerned with finding criteria which permit us to decide the determinacy or indeterminacy of the Hamburger Proving special functions inequalities can be a tedious task and/or Stieltjes moment problems associated with the re- requiring (a combination of) several different techniques. currence relation above on the basis of the behavior of the Among these techniques, algorithms for special functions two coefficient sequences in the recurrence relation. The deserve increasing attention. We will illustrate the scope prototype for this type of result is the classical theorem of and limitations of existing computer algebra methods for Carleman [4]: The Hamburger moment problem associated proving special functions inequalities for some specific ex- with the recurrence relation above is determined if amples that arose in some recent collaborations with Geno ∞ Nikolov (et al.). −1/2 λn = ∞. n=2 Veronika Pillwein Johannes Kepler University We will survey what has been done along these lines and LInz, Austria present new results as well. We also recall a comparison [email protected] theorem due to Carleman which compares the coefficients in a second three term recurrence relation with those in the recurrence relation above and concludes the determinacy MS31 or indeterminacy of the OPS determined by the second Inequalities and Bounds for Some Cumulative Dis- FA15 Abstracts 59

tribution Functions [email protected]

Bounds on some cumulative distributions functions and MS32 their inverses are discussed. In particular, we present re- cent bounds for incomplete gamma functions and Marcum Ladder Operators for Rationally-Extended Poten- functions (also called central and non-central cumulative tials Connected with Exceptional Orthogonal Poly- gamma distributions) and we extend this type of results nomials and Superintegrability to other probability distributions like the beta distribution (central and noncentral). The inverse of the cumulative I will review results concerning k-step extension of the har- distribution functions (quantile functions) are also impor- monic oscillator and the radial oscillator. These 1D ex- tant functions in statistics; we discuss how monotonically actly solvable systems are related to Hermite and Jacobi convergentinversionmethodscanbeusedbothforthenu- exceptional orthogonal polynomials of type III and allow merical inversion and for obtaining analytical bounds for different types of ladder operators. I will show how lad- these functions. der operators involving no isolated multiplets exist and can be constructed via combinations of Darboux-Crum and Krein-Adler SUSYQM approaches. I will also discuss the Javier Segura application to 2D superintegrable systems and derivation University of Cantabria of their energy spectrum using polynomial algebras and Santander, Spain their finite dimensional unitary representations. I will also [email protected] discuss how 1-step and 2-step extension of the harmonic oscillator are connected with a quantum Hamiltonian in- volving the fourth Painleve transcendent with third order MS31 ladder operators.

Tur´an Type Inequalities for Struve Functions Ian Marquette School of Mathematics and Physics We deduce some Tur´an type inequalities for Struve func- The University of Queensland tions of the first kind by using various methods developed [email protected] in the case of Bessel functions of the first and second kind. New formulas, like Mittag-Leffler expansion, infinite prod- uct representation for Struve functions of the first kind, are MS32 obtained, which may be of independent interest. More- BC1 Lam´e Polynomials over, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second The potential of the BC1 quantum elliptic model is a super- kind. position of two Weierstrass functions with doubling of both periods (two coupling constants). The BC1 elliptic model degenerates to A1 elliptic model characterized by the Lam´e Sanjeev Singh Hamiltonian. It is shown that in the space of BC1 elliptic Indian Institute of Technology Madras invariant, the potential becomes a rational function, while [email protected] the flat space metric becomes a polynomial. The model possesses the hidden sl(2) algebra for arbitrary coupling Arp´´ ad Baricz constants: it is equivalent to sl(2)-quantum top in three Department of Economics, Babes-Bolyai University, different magnetic fields. It is shown that there exist three Cluj-Napoca 400591, Romania. one-parametric families of coupling constants for which a [email protected] finite number of polynomial eigenfunctions (up to a factor) occur. They can be called BC1 Lam´e polynomials, being Saminathan Ponnusamy a generalization of the Lam´e polynomials. Indian Statistical Institute, Chennai Centre, CIT Campus, Taramani, Chennai 600113, India. Alexander Turbiner [email protected] Instituto de Ciencias Nucleares, UNAM Departamento de Gravitaci´on y Teora de Campos [email protected] MS32 Coupling Coefficients for Quantum SU(2) Repre- MS32 2 sentations A Dirac-Dunkl Equation on S and the Bannai-Ito Algebra We study tensor products of infinite dimensional irre- The Dirac-Dunkl operator on the 2-sphere associated to the ducible ∗-representations (not corepresentations) of the 3 Z2 reflection group is considered. Its symmetries are found SU(2) quantum group. Eigenvectors of certain ’almost- and are shown to generate the Bannai-Ito algebra. Repre- central’ self-adjoint elements can be given explicitly in sentations of the Bannai-Ito algebra are constructed using terms of q-hypergeometric orthogonal polynomials. We ladder operators. Eigenfunctions of the spherical Dirac- compute coupling coefficients between different eigenvec- Dunkl operator are obtained using a Cauchy-Kovalevskaia tors corresponding to the same eigenvalue; they turn out extension theorem. These eigenfunctions, which corre- to be q-analogs of Bessel functions. As a result we obtain spond to Dunkl monogenics, are seen to support finite di- several q-integral identities involving q-hypergeometric or- mensional irreducible representations of the BannaiIto al- thogonal polynomials and q-Bessel-type functions. gebra. Wolter Groenevelt Hendrik De Bie Delft Institute of Applied Mathematics Ghent University Technische Universiteit Delft [email protected] 60 FA15 Abstracts

Vincent Genest Publications Centre de recherches math´ematiques Universit´edeMontr´eal Condensing the corpus of mathematical research into [email protected] quickly accessible structured content has been the task of review services in mathematics for a long time. Tradition- Luc Vinet ally done by humans (reviews, classification,...), the growth Centre de recherches mathematiques of research - increasingly dispersed in nonstandard form [email protected] like software - requires adapted scalable tools for content analysis. We introduce new automated tools, like math- aware taggers and semantic formula enrichment, currently MS33 employed at zbMATH. An Overview of the Dynamic Dictionary of Mathe- Olaf Teschke matical Functions (http://ddmf.msr-inria.inria.fr) Zentralblatt MATH [email protected] DDMF is a generated, online, interactive dictionary of spe- cial functions. For each function, it displays:local and asymptotic expansions; recurrences on and closed forms of MS34 the coefficients in those expansions; guaranteed arbitrary- Rigorous Borel Summability Methods and Appli- precision numerical approximations; plots; etc. More terms cations to Integrable Models in expansions or more digits in approximations can be obtained upon request. When relevant, human-readable I will present some key results in rigorous Borel summabil- proofs are also automatically generated and displayed. In ity and their applications, with a special emphasis on two the talk, I will demonstrate the website and present the problems: the Dubrovin conjecture for Painev´eP1anda algorithms used. methods of calculating connection formulae in closed form in integrable models, without resorting to Riemann-Hilbert Fr´ed´eric Chyzak or similar reformulations, solely from the Painlev´eprop- INRIA and Ecole´ Normale Sup´erieuredeLyon erty. [email protected] Ovidiu Costin Department of Mathematics MS33 Rutgers University MathSciNet: Digital Guide to the Mathematical [email protected] Literature

MathSciNet is the database based on Mathematical Re- MS34 views, a 75-year-old reviewing service covering the research Uniform Asymptotics of Orthogonal Polynomials mathematics literature. With the exponential growth in Arising from Coherent States scholarly publications, guides to the literature are increas- ingly important. There are several options available, rang- We study a family of orthogonal polynomials {φn(z)} aris- ing from simple internet searches to advanced databases. ing from nonlinear coherent states in quantum optics. MathSciNet provides authoritative information about au- Based on the three-term recurrence relation only, we ob- thors and publications, as well as abstracts and reviews of tain a uniform asymptotic expansion of φn(z)asthepoly- most of the literature. The talk includes a brief history nomial degree n tends to infinity. Our asymptotic results and discussions of scope and functionality. suggest that the weight function associated with the poly- nomials has an unusual singularity, which has never ap- Edward Dunne peared for orthogonal polynomials in the Askey scheme. American Mathematical Society Our main technique is the Wang and Wong’s difference [email protected] equation method. In addition, the limiting zero distribu- tion of the polynomials φn(z) is provided.

MS33 Dan Dai An Introduction to Recent Algorithms Behind the City University, Hong Kong DDMF [email protected]

The DDMF is built on the observation that many special Weiying Hu functions can be completely specified by a linear differen- City University of Hong Kong tial equation and initial conditions. We will describe how [email protected] this data structure can be used to produce numerical values efficiently, expansions in Chebyshev series or other gener- Xiang-Sheng Wang alized Fourier expansions and also in some cases, continued Southeast Missouri State University fraction expansions. [email protected] Bruno Salvy INRIA, Paris-Rocquencourt MS34 [email protected] Some Recursive Techniques in the Approximation of Special Functions

MS33 Many special functions are solutions of a differential or dif- Semantics, Formula Search, Mathematical Soft- ference equation. An appropriate use of the Green function ware - How the zbMATH Database extends Beyond of a certain part of the equation permits the transformation FA15 Abstracts 61

of the differential or difference equation intro an integral nal Polynomials of Weyl Groups or series equation respectively. Then, from the fixed point The link between the discrete Fourier calculus of the four theorem of Banach we obtain a sequence of functions that s l converges to the given special function. As an illustration, families of special functions, C−, S−, S − and S − func- we derive a new convergent expansion of the Bessel func- tions, and the four families of the induced orthogonal poly- tions. nomials is discussed. The affine Weyl groups corresponding to the root systems of simple Lie algebras are recalled and Jose L. Lopez sign homomorphisms, which allow general explicit descrip- State University of Navarra tion of the orbit functions, are described. The discrete [email protected] Fourier calculus of the four types of orbit functions is per- formedforeachtypeonadifferentsetofpointswiththe Chelo Ferreira, Ester Perez Sinusia weights, labeling the orthogonal functions, chosen for each Universidad de Zaragoza type separately. The four types of orthogonal polynomials, [email protected], [email protected] induced by the four types of orbit functions, inherit the dis- crete orthogonality from the orbit functions. The discrete orthogonality of the polynomials is explicitly formulated MS34 and its application for the development of numerical inte- gration formulas and polynomial interpolation methods is Uniform Asymptotic Approximations for Linear discussed. Differential Equations with a Bounded Uniformity Parameter Jiri Hrivnak Department of physics Typically when one studies uniform asymptotic approxi- Czech Technical University in Prague mations for differential equations, the asymptotics is for a jiri.hrivnak@fjfi.cvut.cz large free parameter, say λ, and the approximations are valid for the differentiation variable, say z, near a critical point. Here we will discuss the opposite case. The differ- MS35 entiation variable is large, and the approximations are sup- Lattices of Any Dimension and Their Refinement posed to hold for the free parameter near a critical value. to Any Density Note that in difference equations this is the typical situa- tion, since we normally study the large n asymptotics. A simple definition of a lattice in a real Euclidean space Rn of dimension n, can be stated as follows: the infinite n Adri B. Olde Daalhuis set of discrete points Λ ∈ R is a lattice provided one has Maxwell Institute and School of Mathematics Λ + Λ = Λ, where Λ + Λ stands for every sum of two points The University of Edinburgh of Λ. [email protected] Lattices in the two dimensional Euclidean plane are known to be of two kinds, the lattice of squares and the lattice of equilateral triangles. The lattices, characterized by their MS35 symmetries, come in two forms each. We identify the four cases by the symbols most often used for the correspond- Generalizations of Generating Functions for ing complex semisimple Lie algebras of rank 2: For square Meixner and Krawtchouk Polynomials lattices we use A1 × A1 and C2 , for triangular lattice we use A2 and G2. In this talk we explain how connection and connection-type relations for Meixner and Krawtchouk polynomials may be Marzena Szajewska derived. Using these relations, we obtain generalizations Institute of Mathematics, University of Bialystok of generating functions for Meixner and Krawtchouk poly- [email protected] nomials. From these generalized generating functions, we develop infinite series expressions using the discrete orthog- onality relations for Meixner and Krawtchouk polynomials. MS35 We will also explain how one may obtain orthogonality re- Orthogonality of Macdonald Polynomials with Uni- lations for these polynomials in the complex plane using tary Parameters Ramanujan’s master theorem. Generalizing previous work with Luc Vinet for the sym- Howard Cohl metric group case, we show that for parameters q and t on National Institute of Standards and Technology the unit circle and subject to a suitable truncation rela- Applied and Computational Mathematics Division tion, the Macdonald polynomials associated with crystal- [email protected] lographic root systems satisfy a finite-dimensional system of discrete orthogonality relations on the Weyl alcove. The Roberto S. Costas-Santos discrete orthogonality weights are positive and their total Dept. Physics and Mathematics Alcal´a de Henares, mass is expressed in product form by means of a finitely Madrid, Spa truncated Aomoto-Ito type q-Selberg sum. This gives rise [email protected] to a unitary q-deformed discrete Fourier involution on the Weyl alcove. For q = t our q-deformed Fourier transform Wenqing Xu amounts to the discrete Fourier transform associated with Montgomery Blair High School the discrete orthogonality relations for the Weyl characters [email protected] studied by Kirillov Jr., Korff and Stroppel, and also in a more general form by Patera et al. This is joint work with Erdal Emsiz. MS35 Jan Felipe van Diejen Discrete Orthogonality of Four Types of Orthogo- Instituto de Matem´atica y Fsica 62 FA15 Abstracts

Universidad de Talca In this capacity, PSWFs provide a natural tool for deal- [email protected] ing with bandlimited functions defined on an interval, as demonstrated by Slepian et. al. in a sequence of classical papers. (A function f : R → R is called bandlimited with MS36 band limit c>0 if its Fourier transform is supported on Precise and Fast Computation of Elliptic Functions the interval [−c, c].) Starting with the papers by Slepian et. and Elliptic Integrals al., PSWFs have been used as a tool in electrical engineer- New methods are developed to compute three Jaco- ing (design of antenna patterns), digital signal processing bian elliptic functions, complete and incomplete elliptic (design of digital filters, such as upsampling/downsampling integrals of all three kinds, and their derivatives and algorithms in acoustics), physics (various wave phenomena, inversions by the half and double argument formulas. fluid dynamics, uncertainty principles in quantum mechan- The new methods are of at least 50 bit accuracy and run ics), etc. However, the use of PSWFs has been somewhat 1.1-3.5 times faster than the existing methods: Cody’s Chebyshev approximations, Bulirsch’s cel and el1,and crippled by their slightly mysterious reputation as being ”difficult to compute”. This seems to be related to the Carlson’s RF , RD,andRJ . All the published articles and accompanied Fortran programs are available from fact that the classical (”Bouwkamp”) algorithm for their https://www.researchgate.net/profile/Toshio Fukushima/evaluation encounters numerical difficulties for c>40 or so. Moreover, the attempt to diagonalize the operator Fc numerically via straightforward discretization meets with Toshio Fukushima numerical difficulties as well. In this talk, we describe sev- National Astronomical Observatory of Japan [email protected] eral numerical algorithms for the evaluation of PSWFs, some associated quantities, and PSWFs-based quadrature rules for the integration of bandlimited functions. While MS36 the underlying analysis is somewhat involved, the resulting Computation and Inversion of Certain Cumulative numerical schemes are quite simple and efficient in practi- cal computations, even for large values of band limit (e.g. Distribution Functions 6 c =10 ). Both the direct computation and the inversion of the cumu- lative central beta and gamma distributions (central and Andrei Osipov noncentral) are used in many problems in statistics, ap- Mathematics Department plied probability and engineering. Reliable and fast al- Yale University gorithms for the inversion and computation of these dis- [email protected] tribution functions will be presented. Also, we will show comparisons with other existing algorithms implemented in software platforms like Matlab, Mathematica and R. In MS36 our algorithms, the computation of the cumulative distri- A Fast Chebyshev-Legendre Transform Using An bution functions is based on the use of different methods of Asymptotic Formula approximation such as Taylor expansions, continued frac- tions, uniform asymptotic expansions, numerical quadra- Legendre expansions have applications throughout scien- ture, etc depending on the parameter values. For the in- 2 tific computing because of their L -orthogonality, rapidly version, asymptotic expansions in combination with high- decaying Cauchy transform, and association with spher- order Newton or secant methods are used. ical harmonics. However, fast algorithms for comput- Amparo Gil ing with Legendre expansions are not readily available. Depto. de Matematicas, Est. y Comput. In this talk we describe a fast and numerically stable O 2 Universidad de Cantabria (N(log N) / log log N) algorithm for converting between [email protected] Legendre and Chebyshev expansions based on carefully ex- ploiting an asymptotic formula. Applications are: fast L2- projection, mollification, and the effective computation of Javier Segura Painlev´e transcendents. Departamento de Matematicas, Estadistica y Computacion Universidad de Cantabria, Spain Alex Townsend [email protected] Department of Mathematics MIT [email protected] Nico M. Temme CWI, Amsterdam [email protected] Nick Hale Stellenbosch University [email protected] MS36 On the Evaluation of Prolate Spheroidal Wave Functions and Some Associated Quantities MS37 The First Szego’s Limit Theorem with Varying Co- Prolate spheroidal wave functions (PSWFs) corresponding efficients to band limit c>0 are the eigenfunctions of the truncated 2 − → 2 − Fourier transform Fc : L [ 1, 1] L [ 1, 1] defined via In this talk, I will present some extensions of Szeg¨o’s First the formula Limit Theorem (SFLT) to a class of non Toeplitz matrices.  1 icxt Our results extend those of Kuijlaars and Van Assche for Fc [σ](x)= σ(t) · e dt. (2) −1 Jacobi matrices, as well as those of Tilli on locally Toeplitz FA15 Abstracts 63

matrices. a part of the Saff-Varga Width Conjecture concerning the zero-free regions of these partial sums. Alain Bourget Cal State University, Fullerton Antonio R. Vargas Department of Mathematics Dalhousie University [email protected] Department of Mathematics and Statistics [email protected]

MS37 The Second Szego’s Limit Theorem for a Class of MS38 non-Toeplitz Matrices Universality of Mesoscopic Fluctuations in Orthog- onal Polynomial Ensembles Let a(t) be an integrable function on the unit circle with Fourier coefficients We shall discuss fluctuations on the mesoscopic scale for orthogonal polynomial ensembles and show that these are  π 1 −ikt universal in the sense that two measures with asymptotic ak = a(t) e dt, k ∈Z. 2π −π recurrence coefficients have the same asymptotic meso- scopic fluctuations (under an additional assumption on the The Toeplitz matrices Tn(a) associated with a are local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of

Tn(a)={aj−k}0≤j,k≤n−1 . scales on which the limiting fluctuations are identical. A particular consequence of our results is a Central Limit Szeg˝o’s First Limit Theorem (SFLT) states that for any Theorem for the modified Jacobi Unitary Ensembles on all self-adjoint Toeplitz operator T (a) and any continuous ϕ, mesoscopic scales. This is joint work with Maurice Duits.

 π Jonathan Breuer n Tr[ϕ(Tn(a))] = ϕ(a(t)) dt + o(n). Hebrew University of Jerusalem, Israel 2π −π [email protected] Almost 40 years after establishing the FSLT, Szeg˝oestab- lished the second Szeg˝o’s limit theorem which gives a for- MS38 mula for the o(n) error term above up to exponentially Orthogonal Dirichlet Polynomials small terms. The SFLT has recently been extended to a class of matrices whose diagonal entries satisfy a small de- We discuss orthogonal ”polynomials” formed from linear viation condition. In this talk I will present results toward combinations of Dirichlet functions, as used in L series in extending the second limit theorem to this class of non- number theory. Although formed from the same basis func- Toeplitz matrices. tions as used in the continuous orthogonality of Krein sys- tems, they are distinct from Krein systems. We motivate Tyler McMillen their study, and present some recent results. California State University, Fullerton Department of Mathematics Doron S. Lubinsky [email protected] Georgia Institute of Technology [email protected] MS37 Ratio Asymptotics and Weak Asymptotic Mea- MS38 sures Analytic Continuation of S-property for Multiple Orthogonal Polynomials We will consider the asymptotics of general orthogonal polynomials in the complex plane. Our focus will be on We are interested in the S-property for multiple orthogonal ratio asymptotics and weak asymptotic measures and how polynomials. As a case study, we analyze a family of mul- they relate to the right limits of the Bergman Shift matrix. tiple orthogonal polynomials arising in the normal matrix We will also discuss relative forms of these asymptotics model with cubic + linear potential. In order to perform and generalize some results from the setting of OPUC to analytic continuation of the S-property, we interpret it in measures with more general support. terms of a quadratic differential on the associated spectral curve, and develop a deformation argument on its critical Brian Simanek graph. Vanderbilt University [email protected] Guilherme Silva Katholieke Universiteit Leuven, Belgium [email protected] MS37 Newman-Rivlin Asymptotics for Partial Sums of Pavel Bleher Power Series Department of Mathematics, IUPUI [email protected] We discuss analogues of Newman and Rivlin’s formula con- cerning the ratio of a partial sum of a power series to its limit function and present a new general result of this type MS38 for entire functions with a certain asymptotic character. Nuttall’s Theorem on Algebraic S-contours The main tool used in the proof is a Riemann-Hilbert for- mulation for the partial sums introduced by Kriecherbauer Given a function f holomorphic at infinity, the n-th diago- et al. This new result makes some progress on verifying nal Pad´e approximant to f,say[n/n]f , is a rational func- 64 FA15 Abstracts

tion of type (n, n) that has the highest order of contact feature that one of its generators is purely discrete. It is with f at infinity. Nuttall’s theorem provides an asymp- known that the largedegree asymptotics of such polyno- totic formula for the error of approximation f − [n/n]f in mials is governed by the solution of a vector equilibrium thecasewheref is the Cauchy integral of a smooth den- problem, which was previously computed by V. Sorokin. sity with respect to the arcsine distribution on [−1, 1]. I For the strong asymptotics we use the Riemann–Hilbert will present an extension of Nuttall’s theorem to Cauchy characterization of the Hermite–Pad´e polynomials and the integrals on the so-called algebraic S-contours. corresponding nonlinear steepest descent method. We dis- cuss some of the main ingredients of this analysis and the Maxim Yattselev asymptotic results obtained by this method. This is a joint Indiana University-Purdue University Indianapolis work with A. Aptekarev and G. L´opez-Lagomasino [email protected] Andrei Martinez-Finkelshtein University of Almeria MS39 [email protected] Geometry of Hermite-Pad´e Approximants for a Pair of Cauchy Transforms with Interlacing Sym- metric Supports MS39 On the Convergence of Mixed Type Hermite-Pad´e We consider the multiple orthogonal polynomials with re- Approximants spect to smooth complex measures supported on the real line. In this talk we are interested in the case where the The convergence of diagonal sequences of type II Hermite- supports form two interlacing symmetric intervals and the Pad´e approximants of Nikishin systems have been known ratio of the measures extends to a holomorphic function for some time and recently similar results have been ob- in a region that depends on the size of interlacing. This tained for type I Hermite-Pad´e approximants. In this talk problem was posed and studied by Herbert Stahl in the we present new results on the convergence of diagonal se- 80’s. We shall speak about algebraic functions (of genus 1 quences of a certain mixed type Hermite-Pad´e approxi- and 2) and their abelian integrals (with purely imaginary mation problem of a Nikishin system, which is motivated periods) which define the main term of the asymptotics for in finding approximating solutions of a Degasperis-Procesi this problem. This is joint work with Walter Van Assche peakons problem and in the study of the inverse spectral problem for the discrete cubic string. and Maxim L. Yattselev. Sergio M. Medina Peralta Alexander I. Aptekarev Universidad Carlos III de Madrid Keldysh Institute of Applied Mathematics [email protected] Russian Academy of Science [email protected] Guillermo L´opez Lagomasino UC3M MS39 [email protected] Difference Operators on Lattices and Multiple Or- thogonal Polynomials MS40 We construct difference operators on Z2 using multiple or- Singular Linear Statistics of the Laguerre Unitary thogonal polynomials. Let us stress that it is not clear Ensemble and Painleve III: Double Scaling Analy- whether the eigenvalue problem for a difference equation on sis Z2 has a solution and, especially, whether the entries of an We compute the Hankel determinant generated from a eigenvector can be chosen to be polynomials in the spectral α − − variable. However, for the difference operators in question, singularly deformed weight, w(x; t, α):=x exp( x t/x),x>0,t > 0, in a double scaling scheme, namely, the existence of a polynomial solution to the eigenvalue →∞ → problem is guaranteed if the coefficients of the difference n and t 0, such that s =(2n +1+α)t is fi- operators satisfy a certain discrete zero curvature condi- nite. Asymptotic expansions of the double-scaled determi- tion. In turn, this means that there is a discrete integrable nant are obtained for large and small s. These are found system behind the scene and the discrete integrable system through solutions of a particular Painleve III obtained from can be thought of as a generalization of what is known as the original finite n PIII (with a larger number of param- the discrete time Toda equation, which appeared for the eters.) first time as the Frobenius identity for the elements of the Yang Chen Pade table. University of Macau Maxim Derevyagin [email protected] University of Mississippi USA MS40 [email protected] Christoffel-Darboux-type Formulae for Orthonor- mal Rational Functions and Asymptotics MS39 Weak and Strong Asymptotics for the Pollaczek Consider the reproducing kernel Kn(x, y)= n−1 { }n−1 Multiple Orthogonal Polynomials k=0 ϕk(x)ϕk(y), where ϕk k=0 forms an orthonormal basis for the space of rational functions with poles among Pollaczek multiple orthogonal polynomials are type II {α1,α2,...,αn−1}⊂C ∪{∞}. In the first part of the Hermite–Pad´e polynomials orthogonal with respect to two talk we present Christoffel-Darboux-type formulae for Kn simple measures supported on the positive semi–axis. by exploiting its relation with so-called quasi-orthogonal These measures form a socalled Nikishin pair, with the rational functions. In the second part we use these FA15 Abstracts 65

formulae to obtain asymptotics for the reproducing kernel Kn for n →∞.

Karl Deckers University of Lille, France [email protected]

MS40 Recent Asymptotic Expansions for Legendre Poly- nomial Expansions and Gauss-Legendre Quadra- ture In this talk, we discuss some recent asymptotic expansions related to problems in approximation theory and numerical quadrature. • We present a full asymptotic expansion, as n →∞, of the Legendre polynomial Pn(x)for|x|≤1 − ,  ∈ (0, 1). • We present full asymptotic expansions, as n → ∞, of Legendre series coefficients an =(n +  1 1/2) −1 f(x)Pn(x)dx,whenf(x) has arbitrary algebraic-logarithmic (interior and/or endpoint) sin- gularities in [−1, 1]. • We present a full asymptotic expansion (as the number of abscissas tends to infinity) for Gauss– n Legendre quadrature formula i=1 wnif(xni)forin-  1 tegrals −1 f(x)dx,wheref(x) is allowed to have ar- bitrary algebraic-logarithmic endpoint singularities.

Avram Sidi Computer Science Department Technion - Israel Institute of Technology [email protected]