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Orthogonal Polynomials and Special Functions

SIAM Activity Group on Orthogonal Polynomials and Special Functions

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Published Three Times a Year June 1996 Volume 6, Number 3

Contents of members of our Activity Group. In particular, both the Chair, Charles Dunkl, and the Vice From the Editor ...... 1 Chair, Tom H. Koornwinder, now have their own Memorial Note about Waleed Al-Salam ...... 2 WWW home pages whose addresses are listed on Memorial Note about Lawrence C. Biedenharn . . 3 page 19. Reports from Meetings and Conferences ...... 4 Forthcoming Meetings and Conferences ...... 6 I invite everybody to consult some of the Books and Journals ...... 12 WWW pages which can give interesting infor- Software Announcements ...... 15 Problems and Solutions ...... 15 mation about orthogonal polynomials and special Miscellaneous ...... 18 functions. Rather than looking a formula up in a How to Contribute to the Newsletter ...... 18 mathematical dictionary, the desired information Activity Group: Addresses ...... 19 could be online available to you!

Steven Finch is developing a WWW page on L P O N A L Y Favorite Mathematical Constants and their inter- O N G O O M relations which was announced in the last issue H I A T of the Newsletter. His site includes a wealth of L R SIAM

S O mathematical knowledge and can be visited at Activity Group http://www.mathsoft.com/cgi-shl/constant.bat. S S P E Est. 1990 N C O At http://www.cecm.sfu.ca/projects/ISC/ you I I A C T L F U N have access to the Inverse Symbolic Calculator that looks up in a huge database to identify decimal numbers and uses modern algorithmic methods to generate algebraic identities between sev- From the Editor eral of them. Using these methods, the formula

µ ¶ µ ¶ The electronic media, and, in particular, the X∞ 4 2 1 1 1 k π = − − − World Wide Web, influence the daily work of re- 8k + 1 8k + 4 8k + 5 8k + 6 16 searchers all over the world more and more. This k=0 is reflected, for example, in the continually in- was discovered recently; it gives the opportunity creasing number of personal WWW home pages to calculate far out coefficients of π’s hexadecimal June 1996 Orthogonal Polynomials and Special Functions Newsletter 2

SIAM Activity Group Wilson scheme, equipped with any standardiza- on tion you may wish to use; see also p. 15. Im- Orthogonal Polynomials and Special Functions plementations of the editor are used for this 3 purpose. This interface can be visited at Elected Officers http://www.can.nl/~renes/CAOP.html. Charles Dunkl, Chair You see there is a lot to discover! Strolling Tom H. Koornwinder, Vice Chair through these pages will furthermore link you to Willard Miller, Program Director many other interesting sites. I hope you all can Nico M. Temme, Secretary take advantage of the possibilities which these new Appointed Officers media offer! Wolfram Koepf, Editor of the Newsletter If you know about interesting WWW pages con- Martin E. Muldoon, Webmaster cerning orthogonal polynomials and special func- 3 tions that I did not mention, please submit the The purpose of the Activity Group is corresponding addresses to the Editor. Our mem- —to promote basic research in orthogonal polyno- bers should receive such information. mials and special functions; to further the application of this subject in other parts of mathematics, and in May 31, 1996 Wolfram Koepf science and industry; and to encourage and support the exchange of information, ideas, and techniques between workers in this field, and other mathematicians and scientists. Memorial Note about Waleed Al-Salam 1926-1996 Friends and colleagues are saddened by news of expansion without the computation of early ones, the passing, a few months short of his 70th birth- using single precision arithmetic! day, of Waleed Al-Salam, Professor Emeritus of Mathematics at the University of Alberta. Born Victor Adamchik and Stan Wagon present a on July 15, 1926 in Baghdad, Iraq, he died on deduction of a generalization of the above formula April 14, 1996 in Edmonton, Alberta, Canada. using symbolic computation on their WWW page http://www.wri.com/~victor/articles/pi/pi.html. Waleed studied at the University of California, Berkeley, earning a Bachelor’s degree in Engineer- David H. Bailey and Simon Plouffe who ing Physics in 1950 and a M.A. in Mathematics have discovered the above representation to- in 1951. He returned to Baghdad as an Instructor gether with Peter Borwein have their own at the College of Science for a few years, before WWW page on Recognizing Real Numbers at enrolling for the Ph.D. at Duke University. He http://www.cecm.sfu.ca/organics/papers/bailey/. completed the degree in 1958 with a thesis On the An archive on Numbers with details about Bessel Polynomials written under the supervision history, theory, applications, as well as printed of Leonard Carlitz. By this time he was already and electronic references can be found at a regular contributor to the periodical literature http://archives.math.utk.edu/subjects/numbers.html with some 20 published articles on a variety of containing e.g., a link to continued fractions. topics in orthogonal polynomials and special func- Concerning families of orthogonal polynomi- tions. als, René Swarttouw designed an interface to After completing his Ph.D., Waleed returned Maple which currently enables the generation to the College of Science in Baghdad as Asso- of plots as well as an online computation ciate Professor. Coming back to North America of recurrence, differential and difference equa- in 1962, he eventually (1966) took a position at tions for orthogonal polynomials of the Askey- the University of Alberta, where he was Professor June 1996 Orthogonal Polynomials and Special Functions Newsletter 3 of Mathematics from 1967 until his retirement in Memorial Note about 1992. Waleed continued to contribute to several Lawrence C. Biedenharn areas related to orthogonal polynomials; his CV lists over 80 articles. Areas covered by his work in- Lawrence C. Biedenharn Jr., 73, a longtime clude characterization theorems (see his survey ar- member of Duke University’s physics faculty and ticle in pp. 1–24 of P. Nevai, ed., Orthogonal Poly- an internationally known researcher in theoretical nomials: Theory and Practice, Kluwer, 1990), physics, died Monday, Feb. 12, in Austin, Texas, Turan expressions, generating functions, summa- after a lengthy battle with cancer. He had made tion formulas, q-analogs, and fractional operators. his home in Texas in recent years. He and his collaborators did much work on various special and generalized systems of orthogonal Biedenharn became the youngest full professor polynomials. The Al-Salam-Carlitz polynomials on the Duke faculty—at age 38—when he was ap- (1965) are still frequently cited; see, e.g., the pa- pointed in 1961. He remained at Duke until 1993, pers of R. Askey and S.K. Suslov in Lett. Math. when he retired as James B. Duke professor of Phys. 29 (1993), 123-132 and J. Phys. A 20 (1993), physics and subsequently moved to the University L693-L698. The Al-Salam-Chihara polynomials of Texas at Austin as adjunct professor. (1976) play an important role in the Askey-Wilson A native of Vicksburg, Miss., he received both scheme of basic hypergeometric orthogonal poly- his bachelor’s degree and his Ph.D. from the Mas- nomials. sachusetts Institute of Technology and served in Waleed supervised the Ph.D. work of Bill All- World War II as a Signal Corps officer and later away (1972) and Mourad Ismail (1974), and con- on the staff of Gen. Douglas MacArthur in Japan. ducted joint work with a variety of people at Al- Serving on the faculties of Yale University and berta and elsewhere. In later years, these in- Rice University before coming to Duke, he pub- cluded Ted Chihara, A. Verma, Mourad Ismail lished six books and hundreds of research articles and Waleed’s wife, Nadhla Al-Salam, also on the in the fields of nuclear physics and later math- faculty at Alberta, and a member of this Activity ematical physics. He also edited the Journal of Group. Mathematical Physics for many years. On his retirement in 1992, Waleed decided to Biedenharn, Holman, and Louck showed how put his expertise and energy at the disposal of the the classical work on ordinary hypergeometric se- orthogonal polynomials community by starting ries is intimately related to the irreducible rep- and maintaining an ftp site for papers in the area. resentations of the compact group SU(2). Simi- This effort prospered and he continued to oversee larly, they found U(n) multiple series generaliza- it until last year when his failing health made it tions of one-variable hypergeometric summation necessary to pass the task to Hans Haubold at the and transformation theorems by comparing two UN Office in Vienna. Waleed had been diagnosed ways of computing the matrix elements of multi- with leukemia in 1993 and this was to reduce his plicity free Wigner and Racah coefficients in U(n). active participation in conferences in subsequent This work was done in the context of the quan- years. Nevertheless, with Nadhla’s constant sup- tum theory of angular momentum and the spe- port, he still managed to travel and old and new cial unitary groups SU(n). This work motivated friends were able to benefit from his knowledge the far-reaching q-analogs that Milne, Gustafson, and enjoy his optimistic and humorous personal- and their co-workers subsequently found. Appli- ity. cations of this later work include unified proofs Martin Muldoon of the Macdonald identities, constant term iden- ([email protected]) tities, multiple q-beta integrals, U(n + 1) and symplectic generalizations of the Bailey Trans- form and Bailey Lemma, classical matrix inver- sion results, numerous classical summation and June 1996 Orthogonal Polynomials and Special Functions Newsletter 4 transformation theorems for one-variable q-series, high computational power of this approach by giving Rogers-Ramanujan identities, and, finally, new in- one-line proofs of properties of the Bernoulli polyno- finite families of identities for sums of squares in mials. classical number theory. His book about Quan- • Luis Verde-Star presented a very general Hopf alge- tum Group Symmetry and q-tensor Algebras, bra structure that he showed to be the underlying jointly written with M.A. Lohe, appeared recently, structure of many analytic methods such as Laplace s. page 13. Biedenharn’s work continues to mo- transforms etc. As a nice application he gave a one-line calculation without induction of the integral tivate much of this recent research in multivari- R ∞ xn e−x dx. able orthogonal polynomials, special functions, 0 and their applications. • Jet Wimp showed how cut operators on Laurent series form a powerful tool for proving hypergeometric Lawrence C. Biedenharn is survived by his wife identities. of 45 years, Sarah; his son John; daughter Sally; and two grandchildren. • Mourad Ismail discussed various properties of the Askey-Wilson operators. Monte Basgall • William Chen showed how elementary identities and a ([email protected]) p(qx)−p(x) clever use of the operator ϑ : p(x) 7→ x yields Stephen C. Milne many basic hypergeometric identities and transforma- ([email protected]) tions. • Daniel Loeb presented ongoing joint research with Gian-Carlo Rota and Alessandro Di Bucchianico on Reports from Meetings and Conferences a basis-free infinite-dimensional umbral calculus. As a first application,R he showed how to calculate integrals of the form xd1 . . . xdn dµ(x), where µ is 1. Umbral Calculus Workshop: MIT, April 22-23, ||x||=1 1 n 1996 an orthogonally invariant measure. A workshop on Umbral Calculus was held at M.I.T. (Cam- • Philip Feinsilver showed results from joint research bridge, U.S.A.) on April 22–23 as an addition to the with RenéSchott on linearization and abelianization Rotafest (the celebration of Gian-Carlo Rota’s 64th birth- of Lie algebras with various applications to hypergeo- day). The workshop was organized by Daniel Loeb, Nigel metric functions. Ray and Alessandro Di Bucchianico with much support • Alessandro Di Bucchianico gave an umbral approach from Rotafest organizer Richard Stanley and secretarial to variance functions of exponential families (joint staff of M.I.T. work with Daniel Loeb). He gave a new proof of the The goal of this workshop was to give an overview of the following theorem of Philip Feinsilver: The variance development since Rota’s seminal papers on the subject in function of a natural exponential family is a polyno- the early seventies and to present new developments. Al- mial of degree at most 2 if and only if the associated though Umbral Calculus is classified under combinatorics Sheffer polynomials are orthogonal. in the AMS classification, it has applications to several fields of mathematics, including special functions. We re- • Jack Freeman showed an algebraic transform method fer to Dynamical Survey 3 of the Electronic Journal of that can be used to find orthogonal polynomials (with Combinatorics for more details. respect to a linear functional) within a certain class. He applied his method to the classes of Sheffer poly- Talks were delivered (in chronological order) by Brian nomials and Chebyshev-like polynomials. Taylor, Marilena Barnabei, Heinrich Niederhausen, Luis Verde-Star, Jet Wimp, George Andrews, Mourad Ismail, The workshop was attended by approximately 40 to 60 Miguel Méndez, William Chen, Daniel Loeb, Ottavio people. D’Antona, Nigel Ray, Philip Feinsilver, Henryk Gzyl, Alessandro Di Bucchianico Alessandro Di Bucchianico, and Jack Freeman. Below we ([email protected]) will briefly summarize those talks related to orthogonal polynomials or special functions. 2. CRM Workshop on Special Functions: CRM, May 21-26, 1996 • Brian Taylor presented joint work with Gian-Carlo Rota on a rigorous foundation of the classical umbral This is a report on the Atelier sur les méthodes calculus of the previous century. Brian illustrated the algébriques et q-fonctions spéciales (Workshop on the June 1996 Orthogonal Polynomials and Special Functions Newsletter 5

Algebraic Methods and q-special Functions), Centre nomials de recherches mathématiques, Université de Montréal, Eric Opdam: Spectral analysis of Hecke algebras Montréal,Québec, Canada, May 21-26, 1996. Siddhartha Sahi: Recent results on Jack polynomials and Monday, May 20, was a national holiday in Canada Macdonald polynomials (Victoria Day) and this pushed the start of the workshop to Tuesday morning. Luc Vinet, co-organizer with Pavel Dennis Stanton: q-orthogonal polynomials as moments Winternitz and director of the Centre, opened the session Sergei Suslov (talk delivered by R. Askey): Some basic at 9:00, welcomed the participants and dedicated the work- hypergeometric series and q-Bessel functions. shop to the memory of Waleed Al-Salam, who passed away April 14 of this year (see p. 2 for an obituary). There were Contributed talks were given by N. Atakishiyev, approximately seventeen invited speakers who gave hour- R. Chouikha, P. Floris, A. Grünbaum, K. Kadell, long talks; there were about fourteen contributed half-hour M. Kapilevich, J. LeTourneux, K. Mimachi, A. Odzijew- talks. David and Gregory Chudnovsky were invited but icz, V. Spiridonov, A. Strasburger, N. Takayama, F. van were unable to attend and their time on the program was Diejen, L. Vinet. Ian Macdonald started his lecture Tues- taken by other events. Sergei Suslov was invited, could not day morning and finished it on Wednesday. Dick Askey attend, but Dick Askey delivered his lecture. Dick also gave an extra half-hour talk on Wednesday containing gave his own lecture (more details later). There was in- an overview of Askey-Wilson polynomials. Tom Koorn- deed a heavy emphasis on q-special functions, both of one- winder used another one of the hours originally sched- variable Askey-Wilson type and of several-variable Mac- uled for the Chudnovsky’s to discuss RenéSwarttouw’s donald type. The algebraic methods were highly refined web site which gives access to a vast collection of formu- and sophisticated, mostly based on root systems and as- las and references for the polynomials contained in the sociated mathematical objects such as double affine Hecke Askey tableaux (q = 1 and general q). The URL is algebras. Here is an alphabetical list of the invited speak- http://www.can.nl/˜renes/index.html, see also the soft- ers and the titles of their lectures: ware announcement on p. 15. George Andrews: Plane partitions and MacMahon’s par- There were approximately sixty participants; as well tition analysis some members of the Centre dropped in on the lectures. The languages spoken at coffee and in the hallways ap- Richard Askey: An inequality of Vietoris and some related peared to be English, Russian, French, Japanese, Dutch, hypergeometric sums Polish. The weather was mostly delightful with a few Ivan Cherednik: Spherical difference Fourier transform evening showers. The University is in a scenic location near the Mont Royal; the Pavillon André-Aisenstadt, which Charles Dunkl: Intertwining operators and polynomials houses the Centre, is a beautifully designed and equipped associated with the symmetric group academic building, with a wonderful view of the northwest Pavel Etingof: Macdonald eigenvalue problem and repre- of the city. Ian Macdonald gave the first lecture on Tues- sentations of quantum gl(n) day, Adriano Garsia gave the last one on Saturday, thus bracketing an intense period of leading-edge mathematics. Roberto Floreanini: Quantum algebras and generalized hy- It was generally agreed that the workshop was excellent pergeometric functions both in organization and inspiration to the participants for Adriano Garsia: Polynomiality of the Macdonald q, t- future work. At the conclusion all applauded and thanked Kostka coefficients: a short proof the organizers for this exciting conference. Mourad Ismail: Moment problems and orthogonal polyno- Charles F. Dunkl mials ([email protected])

Tom Koornwinder: The A1-tableau of Dunkl-Cherednik operators 3. Special Functions at the AMS-BeNeLux Meet- ing: Antwerp, Belgium, May 22-24, 1996 Boris Kupershmidt: The great powers of q-calculus From May 22 to 24, the Americam Mathematical Society Ian Macdonald: Symmetric and non-symmetric orthogonal and the Mathematical Societies of the Benelux countries polynomials (Belgium, The Netherlands, and Luxemburg) had a joint David Masson: Contiguous relations, continued fractions meeting in Antwerp, Belgium. The meeting was rather and orthogonality: a ten year journey up the Askey chart successful, with about 500 people attending the confer- Willard Miller, Jr.: Tensor products of q-superalgebras and ence. In the mornings there were plenary speakers. Special q-series identities sessions and contributed papers were planned during the afternoons, with about 12 parallel sessions each afternoon. Masatoshi Noumi: Raising operators for Macdonald poly- June 1996 Orthogonal Polynomials and Special Functions Newsletter 6

There was a special session on wednesday May 22 on an inspiring source of obtaining new relations between hy- Special Functions, organized by Marcel G. de Bruin and pergeometric series, in particular the double hypergeomet- 0:3;3 Walter Van Assche. About 20 people attended this special ric Kampéde Férietseries F1:1;1 of unit argument. Next session. The first two talks were on Sobolev orthogonal Michael Schlosser presented multidimensional matrix in- polynomials. Marcel de Bruin talked about continuous versions and Ar and Dr basic hypergeometric series (joint Sobolev-Laguerre polynomials and their generalized con- work with Ch. Krattenthaler). They computed the inverse tinued fractions. He looked at orthogonal polynomials for of a specific infinite r-dimensional matrix, which they then the inner product applied to obtain many new identities for multidimensional hypergeometric series. Z ∞ hf, gi = f(x)g(x)xαe−x dx Finally the remaining three talks were on computer al- 0 Z ∞ gebra and orthogonal polynomials. Our newsletter editor + λ f 0(x)g0(x)xαe−x dx. Wolfram Koepf gave two talks. The first dealt with effi- 0 cient computation of orthogonal polynomials in computer algebra. He mentioned various ways to compute orthogo- The monic orthogonal polynomials satisfy a four-term re- nal polynomials (explicit series, generating functions, Ro- currence relation, and from this Marcel de Bruin stud- drigues formula, differential equation, recurrence relation, ied the corresponding generalized continued fraction and determinants) and showed how each of these methods per- the connection with simultaneous rational approximation. forms on a computer algebra system (Maple, Mathemat- Henk G. Meijer then talked about coherent pairs for ica, Reduce, MuPad), using the Chebyshev polynomials as Sobolev polynomials. Coherent pairs were introduced by a guiding example. Then he presented orthogonal polyno- Iserles et al. in 1991. If (µ , µ ) is a coherent pair of mea- 1 2 mial identities in Maple V.4. Zeilberger’s algorithm gives sures, then the Sobolev orthogonal polynomials for the in- recurrence relations and differential equations for hyper- ner product geometric sums/integrals and is therefore quite useful for Z orthogonal polynomials in the Askey table. Wolfram gave hf, gi = f(x)g(x) dµ1(x) an overview of Maple’s new capabilities. This was demon- Z strated on the computer. Wolfram implementations were 0 0 + λ f (x)g (x) dµ2(x) used by RenéSwarttouw in his computer implementation of the Askey-Wilson scheme. Renéis presently working can be studied in detail starting from the orthogonal poly- on a project of creating an interactive book on orthogonal polynomials in the Askey-Wilson scheme, with fa- nomials for the measure µ1 (or the measure µ2). The pair (µ, µ), where dµ(x) = xαe−x dx on [0, ∞), studied by de cilities for symbolic manipulation of formulas. He first Bruin in the previous talk, is an example of a coherent pair. showed a clickable Askey-Wilson table, where properties Meijer recently succeeded in finding all coherent pairs. He of each family of orthogonal polynomials can be found just showed that in a coherent pair, one of the measures needs by clicking with the mouse on the required family. Also to be classical (Jacobi, Laguerre, or Hermite), and then limit transitions can be found by clicking the relevant ar- the other measure is a perturbation of the classical mea- rows. Renéalso worked on an interface with Maple, allow- sure (addition of a point mass, multiplication or division by ing anyone to generate recurrence relations and differential a polynomial of degree one). Herman Bavinck then talked equations of any system of orthogonal polynomials in the about A new result for Laguerre, Charlier, and Meixner Askey-Wilson table, even after an affine transformation polynomials. He showed that for all x, α ∈ C and rescaling. Finally he showed that the reverse is also under construction: given the coefficients of a three-term Xi recurrence relation, the user will be informed with which s (−α−i−1) (α+j) k Li−k (−x)Lk−j (x) = 0, particular system of orthogonal polynomials one is deal- k=j ing. RenéSwarttouw’s project can be seen on internet at homepage http://www.can.nl/˜renes/index.html. for all i, j, s ∈ N, provided that i > 2s+j, and gave similar Walter Van Assche results for Meixner polynomials and Charlier polynomials. ([email protected]) The next two talks were on multiple hypergeometric series. First Joris Van Der Jeugt talked on Transformation and summation formulas for multiple hypergeometric func- Forthcoming Meetings and Conferences tions. His starting point is the quantum theory of angular 1. International Memorial Conference D.S. Mitri- momentum. He recalled the 3-j and the 6-j coefficients nović:Niˇs,Serbia, June 20-22, 1996 and indicated that they are related to the Hahn polynomials and the Racah polynomials in the Askey table. Then The International Memorial Conference D.S. Mitrinović he looked at the 9-j coefficients, which he considered as will be held in Niˇs,Serbia, Yugoslavia from June 20–22, June 1996 Orthogonal Polynomials and Special Functions Newsletter 7

1996. The conference will be held on the Faculty of Elec- 3. International Workshop on Orthogonal Polyno- tronic Engineering, Beogradska 14, 18000 Niˇs. It covers mials in Mathematical Physics, in Honour of André the topics: Ronveaux: Madrid, June 24-26, 1996 – Approximation Theory Orthogonal polynomials play a crucial role in a lot of areas – Complex Analysis of Mathematics, Physics, etc. – Differential, Integral and Functional Equations The aim of the Workshop is that a limited number of – General Inequalities invited mathematicians and theoretical physicists discuss – Orthogonal Polynomials and Special Functions and review various recent works of the Theory of Orthogo- The accepted papers will be published in three volumes nal Polynomials and its applications to one of the most in- dedicated to Professor Mitrinović. The titles of volumes teresting areas of knowledge: Mathematical Physics. The will be as follows: Workshop will take place at the Escuela PolitécnicaSupe- 1. Recent Progress in Inequalities rior, Universidad Carlos III de Madrid in Leganés,Madrid. 2. Advances in Mathematical Analysis Invited talks: 3. Topics in Mathematics with Applications N. Atakishiev, Difference equations and some of their The approximate number of pages per volume will be 400. solutions The appearance of the first volume is expected ahead of Jesus S. Dehesa, Information theory, quantum entropy the conference, whereas the other two will appear later. and orthogonal polynomials All volumes will be comprised of survey and contribution papers. Survey papers, due to their length, content and Yuri F. Smirnov, The Wigner-Racah algebra for SUq(2) scientific value may be considered as book chapters, and and SUq(1, 1) and q-analogs of Hahn, Kravchuk and Racah therefore, could be of great interest to a wide range of polynomials. The (quasi) exact solvability concept on the mathematicians. Also, the authors are well-known and finite difference equations have made significant contributions to the relevant topics. H.T. Koelink, Addition formulas for q-special functions. On behalf of the Organizing Committee Hecke algebras and q-Krawtchouk polynomials Prof. Gradimir V. Milovanović,Chair A. Aptekarev, Toda-type dynamics for the coefficients of Faculty of Electronic Engineering recurrence relations Department of Mathematics P.O. Box 73, Beogradska 14 R. Koekoek, The q-analogue of the Askey-scheme of hy- 18000 Niˇs,Yugoslavia pergeometric orthogonal polynomials A. Ronveaux, Connection coefficients for orthogonal Gradimir Milovanovic polynomials ([email protected]) The topics to be considered will be:

2. Joint Summer Research Conference on Ran- 1. q-Polynomials and related topics. dom Matrices, Statistical Mechanics, and Painlevé Transcendents: South Hadley, June 23-27, 1996 2. Entropy of polynomials and related physical problems. This conference will take place at the Mount Holyoke Col- lege, South Hadley, Massachusetts, USA. It is chaired 3. Difference equations and their solutions. jointly by Pavel Bleher and Alexander Its, both of Indi- 4. Finite difference equations and the (quasi) exact solv- ana University-Purdue University at Indianapolis. ability concept. The analysis of correlation functions for exactly solv- able quantum models and of the partition functions in the 5. Toda lattices and orthogonal polynomials. theory of random matrices has gradually become an exciting new branch of mathematical physics which has deep 6. Orthogonal polynomials in Mathematical Physics. connections with both the classical theory of special functions and orthogonal polynomials and the modern theory Organizing Committee: M. Alfaro (Univ. de Zaragoza), of quantum groups and topological quantum field theory. R. Alvarez-Nodarse,´ Secretary (Univ. Carlos III), A.G. Garc´ıa (Univ. Carlos III), G. López Lagomasino Further details can be obtained by WWW: (Univ. Carlos III) and F. Marcellán,Chairman (Univ. Car- http://www.ams.org/committee/meetings/src.html los III). Richard Askey R. Alvarez´ Nodarse ([email protected]) ([email protected]) June 1996 Orthogonal Polynomials and Special Functions Newsletter 8

4. Meeting on Symmetries and Integrability of Dif- 7. SIAM Annual Meeting: Kansas City, July 22- ference Equations: Canterbury, July 1-5, 1996 26, 1996 A meeting on Symmetries and Integrability of Difference The 1996 SIAM Annual Meeting will be held on Equations (SIDE) will be held at the University of Kent July 22–26, 1996 in Kansas City, Missouri, USA. at Canterbury from Monday 1st July to Friday 5th July See http://www.siam.org/meetings/an96/an96home.htm 1996. This conference is the successor of the meeting on for further information, or send an email to the same topics held in Esterel, Quebec, Canada in May [email protected]. A major theme is New Tools of Ap- 1994. plied Mathematics. Carl de Boor will receive the von Neu- mann prize at this meeting. The conference is being organised by Professor Peter Clarkson, Institute of Mathematics & Statistics, University In the February issue of the Newsletter, on p. 7, we wrote of Kent ([email protected]) and Dr. Frank Nijhoff, that our Activity Group probably would not sponsor or Department of Applied Mathematical Studies, Univer- organize any Minisymposium during this Annual Meeting. sity of Leeds ([email protected]). A second an- Fortunately, some proposal has still come up: nouncement, with a preliminary list of speakers and in- The SIAM Activity Group on Orthogonal Polynomials formation about registration is now available at WWW: and Special Functions will present a Minisymposium on http://stork.ukc.ac.uk/IMS/maths/SIDE96/ Modern Topics in Orthogonal Systems. The minisympo- Peter Clarkson sium will be devoted to two modern extensions of the clas- ([email protected]) sical theory of orthogonal systems of special functions in one real variable. The first topic is Wavelets, organized 5. XVth Workshop on Geometric Methods in by Gilbert G. Walter. Talks by Walter, P.R. Massopust, Physics, Bialowieza, Poland, July 1-7, 1996 and T.Q. Nguyen will be given at the first meeting of the minisymposium, 8:30-10:30 on Tuesday morning, July 23, The main emphasis of this workshop will be placed on the and there will be a talk by A. Ron at the second meeting, following topics: Geometric quantization, coherent states, 8:30-10:30, Friday, July 26. The second topic, also treated q-special functions, quantum groups, theory of singulari- at the Friday session, is Multivariate Orthogonal Poly- ties, wavelets, symplectic and Poisson structures, quanti- nomials That Generalize Jacobi Polynomials with Robert zation via ∗-products. Gustafson as the speaker. Both topics are of importance If you are interested to receive further information, in the approximation of functions and representation of please send an email to A. Strasburger (secretary of or- data. Wavelets, in particular, have applications in signal ganizing committee). processing, medicine and biology. Alexander Strasburger Abstracts ([email protected]) Gilbert G. Walter, University of Wisconsin-Milwaukee Improving Wavelet Approximations 6. XXI. International Colloquium on Group The- Abstract: Discrete wavelets are orthogonal sys- oretical Methods in Physics: Goslar, July 15-20, tems on the real line (or in n-space) which 1996 generally have superior convergence proper- Within this Colloquium there will be a.o. a symposium ties compared to classical systems. However, on Quantum Groups covering the following topics: quan- they share a shortcoming—Gibbs’ phenomenon— tum groups and their representations, quantum spaces which causes errors at the edge of a truncated sig- and quantum symmetries, differential calculus on quan- nal or image. It was shown by Shim and Volkmer tum spaces and quantum groups, non-standard deforma- that this always happens for orthogonal approx- tions, Yangians, braided Hopf algebras, relations to non- imations for all continuous wavelets with suffi- commutative Geometry, q-analogues of special functions cient decay. It also occurs for some interpolating and partial differential equations. approximations. Ways of avoiding this for both types of approximations will be presented. The Local Organizing Committee consists of H-D. Doeb- ner, W. Scherer and P. Kramer. For the quantum groups Peter R. Massopust, Sam Houston State Univer- symposium V.K. Dobrev is a co-organizer. sity, Huntsville, Texas. Multiwavelets, Multiresolution Schemes, and Hyperbolic Conservation Laws For up-to-date and additional information see the home Abstract: Multiresolution schemes based on mul- page WWW: http://www.pt.tu-clausthal.de/~group21/ tiwavelets are presented. These schemes employ conference organizers a combination of interpolation and direct eval- ([email protected]) uation and are a generalization of earlier work by A. Harten. It is shown how such multires- June 1996 Orthogonal Polynomials and Special Functions Newsletter 9

olution schemes can be used to obtain accurate a characterization of affine frames; the induced and computationally efficient numerical weak so- characterization of tight affine frames is in terms lutions of partial differential equations arising in of exact orthogonality relations. A general over- computational fluid dynamics. The emphasis is sampling theorem trivially follow from these char- on one-dimensional problems but extensions to acterizations. higher dimensions will be indicated. Numerical Most importantly, the affine product can be fac- experiments are given. tored during a multiresolution analysis construc- Y. Hui, C.W. Kok, T.Q. Nguyen, University of Wiscon- tion, and this leads to a very simple sufficient sin - Madison. Image Coding Using Shift-Invariant Dyadic condition for constructing tight frames from mul- Wavelet Transform tiresolution. Of particular importance are the Abstract: A new class of wavelet filters, shift- facts that the underlying scaling function does invariant wavelet filters, is proposed for the pur- not need to satisfy any a-priori conditions, and pose of image compression. The existing ap- that the freedom offered by redundancy can be proaches obtain the shift-invariant wavelet trans- fully exploited in these constructions. form by finding the path in the decomposi- Willard Miller, Jr. tion tree that minimizes shift-variance with re- ([email protected]) spect to the given cost function. This proce- dure is signal dependent and is inefficient for 8. Seventh International Congress on Computa- image compression, since the subband decompo- tional and Applied Mathematics: Leuven, July 21- sition has to be performed for all shifts of in- 26, 1996 put signal during processing time. The proposed The Congress ICCAM 96 will take place at the Katholieke shift-invariant wavelet transform has better shift- Universiteit Leuven in Belgium and will concentrate on invariant property compared with the conven- the analysis of computational techniques for solving real tional dyadic wavelet transform without changing scientific problems. the structure of it. Two bit allocation schemes, which are suitable for the proposed shift-invariant Invited Speakers: H. Brunner, M.E.H. Ismail, F. Mar- wavelet transform coding, are proposed and eval- cellán,M. Nakao, J. Nedoma, W. Sweldens, P. Toint. uated. Experimental results show that the shift- Please contact M.J. Goovaerts for the full announcement invariant wavelet transform has better energy and an application form. compaction property in image coding compared M.J. Goovaerts to the conventional wavelet transform. ([email protected]) Robert A. Gustafson, Texas A&M University, College Station, Texas. Multivariate Symmetric Orthogonal Poly- 9. AMS-MAA Mathfest: Seattle, Washington, nomials Generalizing Jacobi Polynomials and Interpola- USA, August 10-12, 1996 tion on Symmetric Lattices in Rn Abstract: We discuss a family of multivariate At the AMS-MAA Mathfest (Summer meeting) there will symmetric orthogonal polynomials which have be four lectures dealing with binomial coefficients and the- expansion formulas (as multivariate hypergeo- orems: metric series) and properties similar to classi- – Gian-Carlo Rota: The many lives of binomial coeffi- cal Jacobi polynomials. We also discuss discrete cients. August 12, 10:40 (AMS-MAA invited address) analogs of these polynomials. In the construction – Richard A. Askey: The binomial theorem and some of these orthogonal polynomials, we use a new extensions. family of inhomogeneous symmetric polynomi- Lecture 1: Some of the history of the bino- als, generalizing the classical symmetric functions mial theorem and its extensions (Schur functions). These new symmetric polyno- Lecture 2: Refined counting and a noncom- mials can be used in interpolation problems for mutative version of the binomial theorem symmetric functions on multidimensional symmetric lattices. Lecture 3: Integral analogues of the binomial theorem, orthogonal polynomials and Amos Ron, University of Wisconsin - Madison, education Zuowei Shen, National University of Singapore 10, 11 and 12 August, 9:35 (MAA Earle Raymond Wavelet Frames—a New Approach Hedrick Lectures) Abstract: We unravel the structure of wavelet Tom H. Koornwinder system with the aid of two new notions: the affine ([email protected]) product, and a quasi-affine system. This leads to June 1996 Orthogonal Polynomials and Special Functions Newsletter 10

10. Workshop Transform Methods & Special Func- Further information can be obtained by WWW at tions, II: Varna, August 23-30, 1996 ftp://ecf.hq.eso.org/pub/un/gerannounce.html. The Second International Workshop Transform Methods & Hans J. Haubold Special Functions will be devoted to the 100th Anniversary ([email protected]) of the Bulgarian mathematician Acad. Nikola Obrechkoff (1886–1963). He has left an enormous and valuable her- 12. III. International Conference on Functional itage of more than 250 papers and several monographs Analysis and Approximation Theory: Acquafredda and manuals in various topics: Analysis, Algebra, Num- di Maratea, Italy, September 23-28, 1996 ber Theory, Numerical Analysis, Summation of Divergent The meeting will be devoted to some significant aspects of Series, Probabilities and Statistics etc. New edition and contemporary mathematical research on Functional Anal- translation of the Obrechkoff’s selected papers are also ysis and Approximation Theory including the applications planned. of these fields in other areas. Suggested topics include: The meeting will take place near the town of Varna – Banach spaces, Banach lattices, function spaces (birthplace of N. Obrechkoff), in the Black Sea resort – (Positive) linear operators, semigroups of (positive) Golden Sands, close to the seashore. linear operators, evolution equations Organizing Committee: Prof. Dr. Petar Rusev, Prof. Dr. – Integral equations, interpolation, approximate Ivan Dimovski, Prof. Dr. Shyam L. Kalla (Kuwait Univer- quadratures sity), Asso. Prof. Dr. Virginia Kiryakova, Asso. Prof. Dr. – Approximation methods in abstract spaces and in Lyubomir Boyadjiev. function spaces, approximation by (positive) operators Topics: Integral Transforms, Special Functions, Se- ries Expansions, Fractional Calculus, Algebraical Analy- – Constructive approximation sis, Generalized Functions, Operational Calculus, Univa- – Orthogonal polynomials lent Functions; Applications of these topics to Complex The scientific program will consist of invited survey talks Analysis, Differential and Integral Equations. (45 min.) and short communications (20 min.). The ab- Virginia Kiryakova stracts of all contributions and the program of the meeting ([email protected]) will be available at the beginning of the meeting. The list of invited speakers and a preliminary list of all participants 11. UN/ESA Workshop: Bonn, September 9-13, will be sent together with the second announcement. It is 1996 expected that the proceedings of the Conference will be published. In the past four years the United Nations (UN) in cooper- ation with the European Space Agency (ESA) organized a The meeting is organized in the sphere of activities of series of Workshops on Basic Space Science for the benefit the Center for Studies in Functional Analysis and Approx- of Third World countries in four regions on Earth. imation Theory of the University of Basilicata (Potenza), in collaboration with several other Italian Universities and The Workshop for 1996 is the sixth in the series of Organisations. UN/ESA Workshops on Basic Space Science and will focus on the principal three windows towards the Universe. The organizing committee consists of F. Altomare (Univ. The Workshop will also assess the accomplishments of this of Bari), M. Campiti (Polytechnic of Bari), G. Criscuolo series of Workshops. It will be hosted by Germany, and (Univ. of Napoli), B. Della Vecchia (Univ. of Roma, La will take place at September 9–13, 1996 at the Sapienza), G. Mastroianni (Univ. of Basilicata). Gustav-Stresemann-Institut e.V. The full First Announcement, together with Preliminary Langer Grabenweg 68 Registration Form for participation and for giving a lec- D-53175 Bonn, Germany ture, can be obtained from G. Mastroianni. The completed phone: +49-228-8107-0 form should reach the following address as soon as possible. fax: +49-228-8107-197 Prof. Michele Campiti Dipartimento di Matematica, Universita di Bari Local organizer is the Campus Universitario Max-Planck-Institut fürRadioastronomie Via Edoardo Orabona 4 Auf dem Hügel69 70125 Bari, Italy D-53121 Bonn, Germany email: [email protected] phone.: +49-228-525-0 G. Mastroianni fax: +49-228-525-438 ([email protected]) email: [email protected] June 1996 Orthogonal Polynomials and Special Functions Newsletter 11

13. MSRI Program on Combinatorics: Berkeley. Scope: In recent years there have been many exciting de- Workshops October 14-18, 1996, and April 14-18, velopments in areas that link combinatorics (especially the 1997 theory of symmetric functions) with representation theory and algebraic geometry. This workshop will focus on cur- The MSRI, Berkeley organizes during 1996–97 a full- rent problems in these areas, emphasizing the interplay be- year program on Combinatorics. The program fo- tween algebraic and combinatorial methods. The program cuses on four main areas of approximately half a will include the following topics, as well as others: semester each. During each of these four periods there will be a one-week workshop. See WWW: – Schur functions and their generalizations http://www.msri.org/application/comb.html – Combinatorics and representations of finite Coxeter Two of the areas may have some relevance for the mem- groups bers of our Activity Group: – Quantum groups and Hecke algebras 13.1. Enumeration and partially ordered sets: – Centralizer algebras September 1-October 25, 1996; Workshop: Octo- – Homology representations ber 14-18, 1996 Combinatorial identities (especially computer proofs); con- For more information: jectures on monotone triangles and plane partitions; enu- WWW: http://www.msri.org/sched/CombSymfns.html meration and classification of tilings; combinatorial prob- email: [email protected] lems arising from statistical mechanics and knot theory; Tom H. Koornwinder combinatorial properties of Kazhdan-Lusztig polynomials; ([email protected]) q-analogues and quantum groups. The workshop is being organized by Lynne Butler, Ira 14. Workshop on Special Functions & Differential Gessel, Rodica Simion (chair), and Michelle Wachs. The Equations: Madras, India, January 13-24, 1997 program of the workshop revolves around enumerative and This is a Workshop in the field of Special Functions, Dif- order-theoretic aspects in the study of combinatorial struc- ferential Equations and closely related topics, to be held at tures. Included among the workshop topics are: the Institute of Mathematical Sciences (I.M.Sc.), Madras – q-series during Jan. 13–24, 1997. Mini-series of lectures by experts will introduce the recent trends and developments – Partitions in the topics listed. The objective of the Workshop is to – Plane partitions bring together experts and active research students, to help – Alternating sign matrices strengthening of research activities at the universities and – Enumerative aspects of group algebras and symmetric research institutions in this ever green area. functions Topics include: – Combinatorics of orthogonal polynomials – Special Functions – Computer algebra – Differential Equations – Combinatorial, topological, and algebraic aspects of – Orthogonal Polynomials (One/Several Variables) the theory of partially ordered sets – Group Theory and Special Functions For more information: – Difference Equations WWW: http://www.msri.org/sched/CombPosets.html – q-Special Functions email: [email protected] – Quantum Groups and Special Functions 13.2. Symmetric functions and representation the- – Numerical Methods ory: March 17-May 30, 1997; Workshop: April 14- 18, 1997 – Algebraic/Symbolic Computer Packages Macdonald’s two-parameter symmetric functions and the Plenary speakers include ([*] means: to be confirmed): corresponding two variable Kostka polynomial; immanent R.P. Agarwal (India), B C. Berndt (U.S.A) [*], F. Calogero conjectures of Lieb, Goulden-Jackson, Stembridge, et al.; (Italy), H.D. Doebner (Germany) [*], R.F. Gustafson constant term identities and their connection with nilpo- (U.S.A.) [*], E. Kalnins (New Zealand), T.H. Koornwinder tent Lie algebras and cyclic homology; random walks on (Netherlands), M. Lakshmanan (India), H.L. Manocha (In- groups and shuffling problems; internal products of sym- dia), S.C. Milne (U.S.A.) [*], M. Lohe (Australia), T.D. metric functions; invariant theory. Palev (Bulgaria), C. Quesne (Belgium), A. Ronveaux (Bel- The workshop is being organized by Curtis Greene gium), T.S. Santhanam (U.S.A.), W. Van Assche (Bel- (Chair), Sergey Fomin, Phil Hanlon, and Sheila Sundaram. gium), G. Vanden Berghe (Belgium), J. Van der Jeugt June 1996 Orthogonal Polynomials and Special Functions Newsletter 12

(Belgium), A. Verma (India), L. Vinet (Canada), M. Wald- with the organizer. schmidt (France), P. Winternitz (Canada). Wolfram Koepf There is an international Advisory Committee and a lo- ([email protected]) cal Organizing Committee. Participants should be active research scientists, stu- 16. Continued Fractions and Geometric Function dents, in the field of Special Functions, Differential Equa- Theory: Trondheim, Norway, June 24-28, 1997 tions and related topics. Interested candidates should send Haakon Waadeland celebrates his 70th birthday on 20 May by e-mail (or on plain paper) details giving their name, ad- 1997. He is responsible for a long list of valuable contribu- dress, age, qualifications, present position and a resume of tions to the two fields of continued fractions and geometric research work done. (Maximum number of participants: function theory, and he is still very active. In recognition about 60). of his work we have decided to organize a conference in Financial support for train travel, board and lodging will his honour. The conference will be held in Trondheim, be provided to some of the participants. Registration Fee: Norway from 24 to 28 June 1997 under the title Continued US$ 100 (Rs. 200 for Indian participants) Fractions and Geometric Function Theory. We hope to get together a group of people representing both fields, and we Last date for receiving application: October 14, 1996. would very much appreciate if you would participate. Further information: http://www.imsc.ernet.in/~wssf97/, or by sending a message to: You are all welcome to present talks of 25 minutes and 5 minutes for questions. (If somebody has material suited for Prof. K. Srinivasa Rao longer or shorter presentations, we may be able to arrange Workshop on Special Functions & Diff. Equations this.) We plan to publish proceedings from the conference. The Institute of Mathematical Sciences (I.M.Sc.) The conference has its own world wide web page on CIT Campus, Tharamani http://www.matstat.unit.no/CFGT, where we will put Madras 600113, India new information when it is available. From this page you fax: +91-44-235 0586 can also link to pages about Trondheim and about the K. Srinivasa Rao university, NTNU. ([email protected]) If you want to receive the second announcement, please contact the conference address: [email protected]. 15. First ISAAC Conference: University of Lisa Lorentzen Delaware, Newark, Delaware, June 2-6, 1997 ([email protected]) ISAAC is an abbreviation for The International So- ciety for Analysis, its Applications and Computation. Books and Journals Analysis is understood here in the broadest sense, in- 1. Algebraic Structures and Operator Calculus II: cluding differential equations, integral equations, func- Special Functions and Computer Science, Mathe- tional analysis, and function theory. The first Congress matics and its Applications 292 of this newly constituted society will be held at the By Philip Feinsilver and RenéSchott University of Delaware, between June 2 and June Kluwer Academic Publishers, Dordrecht, The Netherlands, 6, 1997. The conference has its WWW page at July 1994, 160 pp., hardbound, ISBN 0-7923-2021-X, Dfl. http://www.math.udel.edu/isaac/conferen/congr97.htm. 120.00/ US$ 70.00. One section of this conference will be devoted to Or- This is the second of three volumes which present, in an thogonal Polynomials, organized by Wolfram Koepf. The original way, some of the most important tools of applied emphasis of this section will be on the use of symbolic com- mathematics in areas such as probability theory, operator putation in connection with orthogonal polynomials and calculus, representation theory, and special functions, used special functions. Symbolic computation has the potential in solving problems in mathematics, physics and computer to change the daily work of everybody who uses orthogo- science. nal polynomials or special functions in research or appli- This second volume—Special Funtions and Computer cations. It is the purpose of the proposed section to bring Science—presents some applications of special functions in together developers of symbolic algorithms and implemen- computer science. It largely consists of adaptations of arti- tations which are connected with orthogonal polynomials cles that have appeared in the literature, but here they are and special functions with users of computer algebra sys- presented in a format made accessible for the non-expert tems who need this type of software. by providing some context. The material on group repre- This is an early announcement encouraging people who sentation and Young tableaux is introductory in nature. are interested to participate in this section to get in touch The algebraic approach of Chapter 2 is original to the June 1996 Orthogonal Polynomials and Special Functions Newsletter 13 authors and has not appeared previously. Similarly, the 7. Confluent Hypergeometric Functions (includes many material and approach based on Appell states, so formu- special cases) lated, is presented here for the first time. The solutions 8. Legendre Functions are tackled with the help of various analytical techniques, such as generating functions and probabilistic methods and 9. Bessel Functions insights appear regularly. 10. Separating the Wave Equation Contents: Preface · Introduction · Basic Data Structures 11. Special Statistical Distribution Functions (Error func- · Data Structures and Orthogonal Polynomials · Appli- tions, Incomplete Gamma Functions, Incomplete Beta 2 cations of Bessel Functions and Lommel Polynomials · Functions, Non-Central χ Distribution, Incomplete Fourier Transform on Finite Groups and Related Trans- Bessel Function) forms · Young Tableaux and Combinatorial Enumeration 12. Elliptic Integrals and Elliptic Functions in Parallel Processing · References · Index 13. Numerical Aspects of Special Functions (mainly re- Wolfram Koepf currence relations) ([email protected]) Bibliography · Notations and Symbols · Index The author mentions that part of the material was col- 2. Quantum Group Symmetry and q-tensor Alge- bras lected from well-known books such as those by Hochstadt, By L.C. Biedenharn and M.A. Lohe Lebedev, Olver, Rainville, Szeg˝o,and Whittaker & Wat- World Scientific, 1995, x+293 pp. son as well as lecture notes by Lauwerier and Boersma. But there is much recent material, especially in the areas The aim of the monograph is to develop and extend to of asymptotic expansions and numerical aspects. About quantum groups the symmetry techniques familiar from half of the approximately 200 references are dated 1975 the application of classical groups to models in physics. and later. The authors have taken a uniform approach to quantum groups based on the fundamental concept of a tensor op- The author states that the book “has been written with erator. students of mathematics, physics and engineering in mind, and also researchers in these areas who meet special func- The book contains some results about q-special functions in their work, and for whom the results are too scat- tions in connection with quantum groups: in particular tered in the general literature.” Complex analysis (espe- on q-analogues of Clebsch-Gordan coefficients and Racah cially contour integration) would appear to be the main coefficients, and about q-hypergeometric functions in con- prerequisite. The book is clearly written, with good moti- nection with induced representations of the quantized uni- vating examples and exercises. For example, the first chap- versal enveloping algebra for U(3). ter opens with a remarkable example of Borwein, Borwein Tom H. Koornwinder and Dilcher (Amer. Math. Monthly 96 (1989), 681-687) ([email protected]) which explains, using Euler numbers and Boole’s summation formula, why the sum of 50 000 terms of the very 3. Special Functions: An Introduction to the Clas- slowly converging alternating series for π/4, though it gives sical Special Functions of Mathematical Physics an answer correct to only six digits, yet has many correct By Nico M. Temme digits further out in its decimal expansion and, in fact, has Wiley, New York–Chichester–Brisbane–Toronto–Singapore, nearly all its first 50 digits correct. 1996, xii + 374 pp, ISBN 0-471-11313-1, US$ 54.95. Martin Muldoon Contents: ([email protected]) 1. Bernoulli, Euler and Stirling Numbers 4. Index Transforms 2. Useful Methods and Techniques (Theorems from By S.B. Yakubovich Analysis, Asymptotic Expansions of Integrals) World Scientific, Singapore–London–Hong Kong, 1996, 264 pp. 3. The Γ Function This book deals with the theory and some applications 4. Differential Equations (Separating the Wave Equa- of integral transforms that involve integration with re- tion, DE’s in the Complex Plane, Sturm’s Compar- spect to an index or parameter of special functions of ison Theorem, Integrals as Solutions, Liouville Trans- hypergeometric type as the kernel (index transforms). formation) The basic index transforms are considered, such as the Kontorovich-Lebedev transform, the Mehler-Fock trans- 5. Hypergeometric Functions (includes very brief intro- form, the Olevskii transform, the Lebedev-Skalskaya trans- duction to q-functions) forms. The Lp theory of index transforms is discussed and 6. Orthogonal Polynomials new index transforms and convolution constructions are June 1996 Orthogonal Polynomials and Special Functions Newsletter 14 demonstrated. For the first time, the essentially multidi- E.K. Ifantis and P.D. Siafarikas: An alternative proof of a mensional Kontorovich-Lebedev transform is announced. theorem of Stieltjes and related results The book is self-contained, and includes a list of symbols I.H. Jung, K.H. Kwon, D.W. Lee and L.L. Littlejohn: Dif- with definitions, author and subject indices, and an up-to- ferential equations and Sobolev orthogonality date bibliography. S.B. Yakubovich V.A. Kaliaguine: The operator moment problem, vector ([email protected]) continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials 5. Journal of Computational and Applied Mathe- M. Kijima and E.A. van Doorn: Weighted sums of orthog- matics onal polynomials with positive zeros Volume Editor: Marcel G. de Bruin A.B.J. Kuijlaars: Chebyshev quadrature for measures with Volume 65, Dec. 1995, North-Holland, Amsterdam. a strong singularity The Proceedings of the International Conference on Or- S. Lewanowicz: Results on the associated classical orthog- thogonality, Moment Problems and Continued Fractions onal polynomials (dedicated to Thomas Jan Stieltjes Jr.) and held October 31–November 4, 1994 at Delft University of Technology, L. Lorentzen: A convergence question inspired by Stieltjes The Netherlands, have now appeared in this special issue and by value sets in continued fraction theory of the Journal of Computational and Applied Mathemat- A.P. Magnus: Special non uniform lattice (snul) orthogo- ics. We thank the volume editor, Marcel G. de Bruin, for nal polynomials on discrete sets of points supplying us with a LATEX version of the table of contents, from which the text below has been prepared. F. Marcellán,J.C. Petronilho, T.E. Pérezand M.A. Piñar: What is beyond coherent pairs of orthogonal polynomials? Contents G. Mastroianni: Some weighted polynomial inequalities M.G. de Bruin: Preface D.M. Matjila: Bounds for weighted Lebesgue functions for ´ R. Alvarez-Nodarse, A.G. Garc´ıa,and F. Marcellán: On Freud weights on a larger interval the properties for modifications of classical orthogonal polynomials of discrete variables M.E. Muldoon: Electrostatics and zeros of Bessel functions H. Bavinck: The zeros of a certain linear combination of O. Nj˚astad: Extremal solutions of the strong Stieltjes mo- Chebyshev polynomials ment problem C. Berg: Indeterminate moment problems and the theory F. Peherstorfer: Stieltjes polynomials and functions of the of entire functions second kind A. Bultheel, P. González-Vera and R. Orive: On the con- F. Peherstorfer and R. Steinbauer: Characterization of or- vergence of general two-point Padéapproximants to Stielt- thogonal polynomials with respect to a functional jes functions. Part I: Algebraic aspects M. Rösler: Trigonometric convolution structures on Z de- W.C. Connett and A.L. Schwartz: Subsets of R which sup- rived from Jacobi polynomials port hypergroups with polynomial characters A. Sinap: Gaussian quadrature for matrix valued functions H. Dette: On the minimum of the Christoffel function on the real line K. Diethelm: Gaussian quadrature formulae of the third F.H. Szafraniec: A method of localizing the spectra of se- kind for Cauchy principal value integrals: Basic properties quences of orthogonal polynomials and error estimates N.M. Temme: Uniform asymptotic expansions of integrals: J. Dombrowski and S. Pedersen: Orthogonal polynomials, a selection of problems spectral measures and absolute continuity G. Valent and W. Van Assche: The impact of Stieltjes’ K.A. Driver: Nondiagonal quadratic Hermite-Padé ap- work on orthogonal polynomials: additional material proximation to the exponential function M. Voit: Limit theorems for random walks on the double S. Ehrich: Asymptotic behaviour of Stieltjes polynomials coset spaces U(n)/U(n − 1) for n → ∞ for ultraspherical weight functions List of talks presented at the conference D. Fasino: Spectral properties of Hankel matrices and nu- List of registered participants merical solutions of finite moment problems Author Index D.P. Gupta and D.R. Masson: Contiguous relations, con- Martin Muldoon tinued fractions and orthogonality: an 8φ7 model ([email protected]) June 1996 Orthogonal Polynomials and Special Functions Newsletter 15

6. Electronic Journal of Combinatorics: Foata Further information about this project is now available Festschrift from WWW: http://www.can.nl/~renes/index.html Volume Editors: J. Désarménien, A. Kerber, V. Strehl From there you can also download a preliminary ver- Volume 3 (2), 1996. sion of the electronic book version. It is a dvi file named AW.dvi, which is equipped with hyperlinks. This file can A special issue of The Electronic Journal of Combinatorics be read by using the program xhdvi. Consult also the appeared which is dedicated to Dominique Foata on the oc- above WWW site for the way to obtain xhdvi. There is casion of his 60th birthday: another version available which can be read via Netscape. http://ejc.math.gatech.edu:8080/Journal/Volume_3/festschrift.html It uses a dvi viewer created by Garth Dickie. It contains 25 contributions, some of which are in the Swarttouw is also working on a package for calculat- field of orthogonal polynomials or relevant for this field. I ing formulas for orthogonal polynomials belonging to the mention a few and I recommend the reader to inspect the Askey-scheme by Maple. At the mentioned WWW site contents for other titles: there is an online facility to do some of these computa- tions, for which you do not need to have Maple on your R1: Daniel Barsky and Michel Carpentier: Polynômesde own computer. The present demos use Maple procedures Jacobi généraliséset intégralesde Selberg (33pp) written by Wolfram Koepf. R13: Doron Zeilberger: Proof of the alternating sign ma- Tom H. Koornwinder trix conjecture (84pp) ([email protected]) R16: Shalosh B. Ekhad and J.E. Majewicz: A short WZ- style proof of Abel’s identity (1p) Problems and Solutions R17: Dominique Dumont and Armand Ramamonjisoa: Thus far 14 problems have been submitted five of which Grammaire de Ramanujan et arbres de Cayley (18pp) have been solved (#1, 4, 6, 7, 10), and one of which is R19: Marko Petkovˇsekand Herbert S. Wilf: A high-tech new (#14). proof of the Mills-Robbins-Rumsey determinant formula (3pp) Solutions of the problems #11 and #12 are presented in the current issue. R20: Alun Morris and A. A. Abdel-Aziz: Schur Q- functions and spin characters of symmetric groups I 2. Is it true that (14pp) 2 x x t 2F1(x + 1, x + 1; 2; 1 − t) R21: George E. Andrews: Pfaff’s method (III): Compari- is a convex function of x whenever −∞ < x < ∞ and son with the WZ method (18pp) 0 < t < 1? R24: A.M. Garsia and M. Haiman: Some natural bigraded Submitted by George Gasper, August 19, 1992. Sn-modules (60pp) (related to Macdonald polynomials) ([email protected]) Tom H. Koornwinder 3. The following Toeplitz matrix arises in several appli- ([email protected]) cations. Define for i 6= j sin απ(i − j) A (α) = , Software Announcements ij π(i − j)

1. Askey-Wilson Computer Algebra Mini-Project and set Aii = α. Conjecture: the matrix As already announced on p. 13 of the February Newslet- M = (I − A)−1 ter 6.2, René Swarttouw is working during the period January-June 1996 at RIACA, Eindhoven, The Nether- has positive entries. A proof is known for 1/2 ≤ α < 1. lands on a project A computer implementation of the Can one extend this to 0 < α < 1? Askey-Wilson scheme. The purpose of the project is to Submitted by Alberto Grünbaum, November 3, 1992. make a start with bringing the report ([email protected]) R. Koekoek and R.F. Swarttouw The Askey-scheme of hypergeometric orthogonal 5. The result of Problem #4 can be generalized to polynomials and its q-analogue X∞ (−1)n(mn + 1/2)! S = √ Report 94-05, Delft University of Technology, m π (mn + 1)! Faculty TWI, 1994 n=0 1 mX−1 sin (5(2k + 1)π/(4m) + π/4) to the form of an interactive book, with facilities for sym- = m [2 sin ((2k + 1)π/(2m))]1/2 bolic manipulation of formulas. k=0 June 1996 Orthogonal Polynomials and Special Functions Newsletter 16 valid for integral m ≥ 2. 12. Absolutely Monotonic Function. The functions t (x) generated by Submitted by J. Boersma and P.J. de Doelder, n July 12, 1993. X∞ 1 n ([email protected]) ¡ √ ¢α = tn(x) z 2 2 (1 − z) 1 − z + 1 − 2xz + z n=0 8. Can the real and imaginary parts of a hypergeometric series of type pFq with one complex parameter (either in are nonnegative for α ≥ 0, x ∈ [−1, 1]. Can this be proved the numerator or the denominator) be expressed by means by elementary means? of multiple hypergeometric series? Submitted by Dieter Schmersau, January 8, 1996. Submitted by Ernst D. Krupnikov, July 25, 1993. ([email protected]) ([email protected]) 13. Product of Chebyshev Polynomials. For any pair of positive even n, m ∈ N let 9. Prove or disprove: The functions n−1 µ ¶ n Y m ³ (2k + 1)π ´ Hn(t) = (−1) Fn(nt) F (n,m)(x) = 2n cosh arccosh x−cos 2 n k=0 that are defined in terms of the Bateman functions µ ¶ ³ ´ nY−1 ³(2k + 1)π ´ −t = 2n T x − cos , Fn(t) = e Ln(2t) − Ln−1(2t) m/2 n k=0 −t 2t (1) = −e Ln−1(2t) n where Tm(x) denotes the Chebyshev polynomials of the Z π/2 first kind. These functions occur in statistical physics. n 2 = (−1) cos (t tan θ − 2 n θ) dθ , They constitute polynomials in x π 0 nm/4 (α) X Ln (t) denoting the generalized Laguerre polynomials, (n,m) 2j F (x) = Aj(n, m) x have the property that Hn(t0) is strictly decreasing with j=0 increasing n at the point t0 = 2. Note that this is not true for any t0 < 2, but on the other hand at t0 = 2 seems to be whose coefficients Aj(n, m) are integers. Show the sym- numerically evident. Note further that Hn(t) satisfy the metry simple differential equation F (n,m)(x) = F (m,n)(x) ,

00 2 and give a representation of the coeﬃcients A (n, m). t Hn (t) = n (t − 2) Hn(t) . j Submitted by Christian Hege, January 16, 1996. The diﬀerential equation demonstrates the importance of ([email protected]) the point t0 = 2. 14. A Trigonometric Integral. Evaluate the integral These functions occur particularly in the study of nonvanishing analytic functions of the unit disk. For more de- Z π/4 tails on these functions, see Koepf, W. and Schmersau, D.: ln(sin3/2(x) + cos3/2(x)) dx Bounded nonvanishing functions and Bateman functions, 0 Complex Variables 25 (1994), 237–259. which arose in a study [1] dealing with orthogonal polyno- Submitted by Wolfram Koepf, February 10, 1995. mials. ([email protected]) [1] Joyce, G.S., private communication. 11. A Bessel-Laplace Transform. For p ≥ 2 show Submitted by I.J. Zucker, March 8, 1996. that Z ([email protected]) ∞ 1 I = e−ptJ (t)K (at)dt = K(k) 0 0 2p 0 Solution of Problem 11 where q p A Bessel-Laplace Transform a = p2 − 1 ± p p2 − 4 by Lawrence Glasser r ([email protected]) 1 1 1 p k = p2 + 1 ∓ p p2 − 4 . p 2 2 By using the identity [1] Z ∞ Z 1 Z ∞ p Submitted by Lawrence Glasser, Nov 9, 1995. −t 2 e f(t)dt = dy xdxJ0(x 1 − y )f(xy) ([email protected]) 0 0 0 June 1996 Orthogonal Polynomials and Special Functions Newsletter 17

∞ n ∞ n and [2;(8.13.8)] one finds 1 X X X X = · P (α,0)(x) zn · C(−1/2)(x) zn , 2α j j Z ∞ Z 1 n=0 j=0 n=0 j=0 −t dy e J0(at)K0(ct)dt = p 4 2 0 0 Ay + 2By + 1 and the result follows from the positivity of the Jacobi polynomial sums with Xn 2 2 2 2 2 2 (α,0) A = (a + c ) + 2(a − c ) + 1,B = c − a − 1. Pj (x) ≥ 0 (α > −2) (1) j=0 Hence, Z ∞ (see [2], Theorem 3), which is the so-called Askey-Gasper −pt e J0(at)K0(bt)dt = inequality, and the positivity of the Taylor coefficients 0 µ µ µ ¶ ¶¶ (with respect to z) of the function 1 1 α − p2 √ K(k) − F cos−1 , k √ α 2 α + p2 1−2xz+z2 1 − z with p 2 2 2 2 2 2 4 α = (a + b ) + 2p (a − b ) + p , (see [2], Theorem D). r α + a2 − b2 + p2 k = . References 2α [1] Abramowitz, M. and Stegun, I. A.: Handbook of Math- This is a complete elliptic integral when α = p2, e.g. Thus, ematical Functions. Dover Publ., New York, 1964. for (b2 + 1)2 = 2p2(b2 − 1), Z [2] Askey, Richard and Gasper, George: Positive Jacobi ∞ 1 p e−ptJ (t)K (bt)dt = K( 1 − (b2 − 1)/2p2) polynomial sums II. Amer. J. Math. 98 (1976), 709– 0 0 2p 0 737. which is equivalent to the stated result. The same proof was given by Dick Askey who wrote to the References proposer [1] Glasser, M.L. and Kowalenko, V., to be published. I can prove the claim you made in problem 12. [2] Erdélyi,A., Magnus, W., Oberhettinger, F. and Tri- Take the generating function for the Jacobi poly- comi, F.G.: Tables of Integral Transforms, Vol. 2. nomial√ with parameters α and zero, and multi- 2 −2 McGraw-Hill, New York, 1954. ply by 1 − 2xz + z . Then split (1 − z) into two equal parts. One of the factors is an absolutely monotonic function by an easy result of Solution of Problem 12 Askey and Pollard, which does not use anything Absolutely Monotonic Function difficult. The other is what is called the Askey- by Dieter Schmersau Gasper inequality (1). There are many proofs of ([email protected]) this, and what one calls elementary depends on what you know. I can make a case that it is el- To deduce the inequalities tn(x) ≥ 0, we remark that the ementary, but most would not agree with me. Is (α,β) Jacobi polynomials Pn (x) have the generating function this how you proved this result?

∞ George Gasper wrote X 2α+β P (α,β)(x) zn = √ n 1−2xz+z2 You could use the Fields and Wimp expansions n=0 in their Math. of Computation 15 (1961), pp. 1 1 390-395, paper as I did to derive Eq. (8.18) on · ¡ √ ¢α ¡ √ ¢β 1 − z + 1−2xz+z2 1 + z + 1−2xz+z2 p. 415 of my Positivity and Special Functions paper published in the book Theory and Applica- (see e.g. [1], (22.9.1)), hence tion of Special Functions, ed. by R.A. Askey, Aca- 1 demic Press, 1975. ¡ √ ¢α = (1 − z)2 1 − z + 1 − 2xz + z2 Indeed, the only open question remains whether or not the √ Askey-Gasper inequality is elementary or not. 1 1 1 1−2xz+z2 √ ¡ √ ¢α · 1 − z 1−2xz+z2 1−z+ 1−2xz+z2 1 − z June 1996 Orthogonal Polynomials and Special Functions Newsletter 18

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For several years William Connett and I have organized summer Wolfram Koepf research groups designed to give a mathematical experience to Konrad-Zuse-Zentrum fürInformationstechnik (ZIB) undergraduates and high school students. A typical activity Heilbronner Str. 10 would be to use a computer algebra system to test conjectures D-10711 Berlin, Germany about orthogonal polynomials or produce interesting graphics. phone: +49-30-896 04-216 fax: +49-30-896 04-125 We would be very grateful for any suggestions for projects. email: [email protected] Alan L. Schwartz ([email protected]) preferably by email, and in LATEX format. Other formats are also acceptable and can be submitted by email, regular mail or 2. Revising the 1991 Mathematics Subject Classifica- fax. tion Deadline for submissions to be included in the October issue 1996 is September 15, 1996. The following was taken from Notices AMS, December 1995, p. 1547, see also The Activity Group also sponsors an electronic news net, http://www.ams.org/committee/publications/msc-2000-let.html called the OP-SF Net, which is transmitted periodically by I suggest that you send suggestions for revision of num- SIAM. The Net provides a rather fast turnaround compared to bers 33 (Special functions), 42Cxx (Nontrigonometric Fourier the Newsletter. To receive transmissions, just send your name analysis) and 44 (Integral transforms, operational calculus) and email address to [email protected] (as with other also to the editors of OP-SF Net ([email protected] and mul- nets, nonmembers can also receive the transmissions). Your [email protected]) or to the editor of the Newsletter OP-SF Net contributions should be sent to [email protected]. ([email protected]) in order that a summary of suggestions Please note that submissions to OP-SF Net are automatically for this subject can be listed, and maybe discussed by the mem- considered for publication in the Newsletter, and vice versa, bers of the Activity Group. unless the writer requests otherwise. The Net is organized by Tom Koornwinder ([email protected]) Note from Notices AMS: and Martin Muldoon ([email protected]). Back is- Mathematics Subject Classification Scheme Revision sues of OP-SF Net can be obtained by anonymous ftp from To the Mathematical Community: ftp.fwi.uva.nl, in the directory pub/mathematics/reports/Analysis/koornwinder/opsfnet.dir The editors of Mathematical Reviews and Zentralblatt für or by WWW at the addresses Mathematik have initiated the process of revising the 1991 ftp://ftp.fwi.uva.nl/pub/mathematics/reports/Analysis/ Mathematics Subject Classification, which is used by both jour- koornwinder/opsfnet.dir nals as their classification system. The editors do not plan a radical revision of the present 1991 system, but it is clear that http://www.math.ohio-state.edu/JAT some changes will be needed in order to accommodate recent Martin Muldoon, moreover, manages our home page developments in mathematical research. http://www.math.yorku.ca/Who/Faculty/Muldoon/siamopsf/ It will be necessary to have this revision completed by the end on World Wide Web. Here you will find also a WWW version of 1998 so that it can begin to be used in Current Mathematical of the OP-SF Net. It currently covers the topics Publications in mid 1999, and in Mathematical Reviews and Zentralblatt fürMathematik beginning in 2000. – Conference Calendar – Books, Conference Proceedings, etc. We hereby solicit comments and suggestions from the math- – Compendia, tools, etc. ematical community to be considered in this revision process. These should be submitted by June, 1997. The preferred – Meeting Reports method of communication is by e-mail: – Projects [email protected] or msc2000@zblmath.fiz-karlsruhe.de – Problems (Comments and suggestions may also be sent to either one of – Personal, Obituaries, etc. us at the addresses given below.) We are eager that research – Positions available mathematicians and scholars have input in this revision process – Miscellaneous as soon as possible. R. Keith Dennis Bernd Wegner Executive Editor Chefredakteur Mathematical Reviews Zentralblatt fürMathematik Tom H. Koornwinder ([email protected]) June 1996 Orthogonal Polynomials and Special Functions Newsletter 19

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