Volume 18, Number 4 – July 15, 2011
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Abramowitz Function Computed by Clenshaw's Method, 74 Absolute
Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. Index Abramowitz function for parabolic cylinder functions, computed by Clenshaw’s method, 378 74 for prolate spheroidal harmonics, absolute error, 356 364 Airy function for Scorer functions, 361 contour integral for, 166 for toroidal harmonics, 366 Airy functions of Remes, 290 algorithm, 359 analytic continuation of generalized asymptotic estimate of, 18 hypergeometric function, 27 asymptotic expansions, 81, 360 anomalous behavior of recursions, 118 Chebyshev expansions, 80, 85 a warning, 122 computing confluent hypergeometric complex arguments, 359 functions, 120 Gauss quadrature, 145 exponential integrals, 121 scaled functions, 359 first order inhomogeneous zeros, 224 equation, 121 connection formulas, 360, 361 modified Bessel functions, 118 contour integral for, 264 anti-Miller algorithm, 110, 112 differential equation, 249, 359 associated Legendre functions relation with hypergeometric computation for z>0, 363 function, 28 asymptotic expansion used in uniform asymptotic uniform, 237 expansion, 250 asymptotic expansions Airy-type asymptotic expansion alternative asymptotic for modified Bessel functions of representation for (z),49 purely imaginary order, 375 alternative expansion for parabolic cylinder functions, for (z),49 383 for (a, z),47 obtained from integrals, 249, 264 convergent asymptotic algorithm representation, 46 for Airy functions, 359 converging factor, 40 for computing zeros of Bessel exponentially improved, 39 functions, 385 for (a, z),39 for modified Bessel functions, 370 exponentially small remainders, 38 for oblate spheroidal harmonics, hyperasymptotics, 40 365 of exponential integral, 37, 38 405 From "Numerical Methods for Special Functions" by Amparo Gil, Javier Segura, and Nico Temme Copyright ©2007 by the Society for Industrial and Applied Mathematics. -
Large Parameter Cases of the Gauss Hypergeometric Function
1 Large Parameter Cases of the Gauss Hypergeometric Function Nico M. Temme CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands e-mail: [email protected] ABSTRACT We consider the asymptotic behaviour of the Gauss hypergeometric function when several of the parameters a, b, c are large. We indicate which cases are of interest for orthogonal polynomials (Jacobi, but also Krawtchouk, Meixner, etc.), which results are already available and which cases need more attention. We also consider a few examples of 3F2 functions of unit argument, to explain which difficulties arise in these cases, when standard integrals or differential equations are not available. 2000 Mathematics Subject Classification: 33C05, 33C45, 41A60, 30C15, 41A10. Keywords & Phrases: Gauss hypergeometric function, asymptotic expansion. Note: Work carried out under project MAS1.2 Analysis, Asymptotics and Computing. This report has been accepted for publication in the proceedings of The Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications, 18-22 June 2001, Rome, Italy. 1. Introduction The Gauss hypergeometric function ∞ a, b ab a(a + 1) b(b + 1) (a) (b) F ; z =1+ z + z2 + . = n n zn, (1.1) 2 1 c c c(c + 1) 2! (c) n! n=0 n X where Γ(a + n) a (a) = = ( 1)n n! − , (1.2) n Γ(a) − n arXiv:math/0205065v1 [math.CA] 7 May 2002 is defined for z < 1 and c = 0, 1, 2,... | | 6 − − A simplification occurs when b = c: a, b F ; z = (1 z)−a. (1.3) 2 1 b − When a or b are non-positive integers the series in (1.1) will terminate, and F reduces to a polynomial. -
Bibliography
Bibliography 1. N. ABDUL-HALIM and W.A. AL-SALAM, A characterization of the Laguerre polynomials. Rendiconti del Seminario Matematico della UniversitadiPadova` 34, 1964, 176–179. 2. M. ABRAMOWITZ and I.A. STEGUN (eds.), Handbook of mathematical functions (with for- mulas, graphs, and mathematical tables). Dover Publications, New York, 1970. 3. L.D. ABREU and J. BUSTOZ, Turan´ inequalities for symmetric Askey-Wilson polynomials. The Rocky Mountain Journal of Mathematics 30, 2000, 401–409. 4. S. AHMED,A.LAFORGIA and M.E. MULDOON, On the spacing of the zeros of some classi- cal orthogonal polynomials. Journal of the London Mathematical Society (2) 25, 1982, 246– 252. 5. S. AHMED,M.E.MULDOON and R. SPIGLER, Inequalities and numerical bounds for ze- ros of ultraspherical polynomials. SIAM Journal on Mathematical Analysis 17, 1986, 1000– 1007. 6. K. ALLADI and M.L. ROBINSON, Legendre polynomials and irrationality. Journal fur¨ die Reine und Angewandte Mathematik 318, 1980, 137–155. 7. WM.R. ALLAWAY, The representation of orthogonal polynomials in terms of a differential operator. Journal of Mathematical Analysis and Applications 56, 1976, 288–293. 8. WM.R. ALLAWAY, Some properties of the q-Hermite polynomials. Canadian Journal of Mathematics 32, 1980, 686–694. 9. WM.R. ALLAWAY, Convolution orthogonality and the Jacobi polynomials. Canadian Math- ematical Bulletin 32, 1989, 298–308. 10. WM.R. ALLAWAY, Convolution shift, c-orthogonality preserving maps, and the Laguerre polynomials. Journal of Mathematical Analysis and Applications 157, 1991, 284–299. 11. N.A. AL-SALAM, Orthogonal polynomials of hypergeometric type. Duke Mathematical Jour- nal 33, 1966, 109–121. -
Studies on the Generalized and Reverse Generalized Bessel Polynomials
STUDIES ON THE GENERALIZED AND REVERSE GENERALIZED BESSEL POLYNOMIALS ZEYNEP SONAY POLAT APRIL 2004 STUDIES ON THE GENERALIZED AND REVERSE GENERALIZED BESSEL POLYNOMIALS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF THE MIDDLE EAST TECHNICAL UNIVERSITY BY ZEYNEP SONAY POLAT IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF MATHEMATICS APRIL 2004 Approval of the Graduate School of Natural and Applied Sciences Prof. Dr. Canan OZGEN¨ Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science. Prof. Dr. S¸afak ALPAY Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Prof. Dr. Hasan TAS¸ELI˙ Supervisor Examining Committee Members Prof. Dr. M¨unevver TEZER Prof. Dr. Hasan TAS¸ELI˙ Prof. Dr. Marat AKHMET Prof. Dr. A˘gacık ZAFER Dr. Om¨urU¨ GUR˘ I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name : Zeynep Sonay Polat Signature : abstract STUDIES ON THE GENERALIZED AND REVERSE GENERALIZED BESSEL POLYNOMIALS Polat, Zeynep Sonay M.Sc., Department of Mathematics Supervisor: Prof. Dr. Hasan Ta¸seli April 2004, 73 pages The special functions and, particularly, the classical orthogonal polynomials encountered in many branches of applied mathematics and mathematical physics satisfy a second order differential equation, which is known as the equation of the hypergeometric type. -
Address: Education: Academic Experience: Visiting Positions
Lance L. Littlejohn Address: Department of Mathematics Phone (offi ce): 254-710-3943 Sid Richardson Building 303E Baylor University Fax: 254-710-3569 Waco, Texas Email: [email protected] 76798-7328 Webpage: https://www.baylor.edu/math/index.php?id=53980 Education: Institution Dates Attended Degree Date Degree University of Western Ontario 1970-1975 1975 BSc. University of Western Ontario 1975-1976 1976 M.A. Pennsylvania State University 1976-1981 1981 Ph.D. Academic Experience: University of Western Ontario, 1975-1976; Teaching Assistant Pennsylvania State University, 1976-1981; Graduate Assistant The University of Texas at San Antonio, 1981-1983; Assistant Professor Utah State University, 1983-1987; Assistant Professor Utah State University, 1987-1993; Associate Professor Utah State University, 1993-2006; Full Professor Baylor University, 2007-2019; Full Professor and Chair of Department Baylor University, 2012-2016; Associate Dean for Research, Graduate School Baylor University, August 2019 - present; Full Professor Visiting Positions: University of Surrey (England), 1987; Visiting Associate Professor University of South Africa, 1990; Visiting Associate Professor University of Wales, Cardiff, 1990-1991; Visiting Associate Professor Carlos III University, Madrid, Spain, 1999-2000; Visiting Full Professor Carlos III University, Madrid, Spain, 2002; Visiting Full Professor University of Gabes, Tunisia, Visiting Full Professor KAIST, Daejeon, South Korea, 2002-2004; Visiting Full Professor 1 Refereed Publications: 1982: 1. “Nonclassical orthogonal polynomials as solutions to second order differential equa- tions”, (with S.D. Shore), Canad. Math. Bull., 25(3), 1982, 291-295. 2. “The Krall polynomials: a new class of orthogonal polynomials”, Quaestiones Math., 5(3), 1982, 255-265. 1983: 3. “A singular sixth order differential equation with orthogonal polynomial eigenfunc- tions”, (with A.M. -
Linearization Coefficients of Bessel Polynomials and Properties Of
Linearization coefficients of Bessel polynomials and properties of Student-t distributions Christian Berg and Christophe Vignat August 5, 2005 Abstract We prove positivity results about linearization and connection coeffi- cients for Bessel polynomials. The proof is based on a recursion formula and explicit formulas for the coefficients in special cases. The result implies that the distribution of a convex combination of independent Student- t random variables with arbitrary odd degrees of freedom has a density which is a convex combination of certain Student-t densities with odd degrees of freedom. 2000 Mathematics Subject Classification: primary 33C10; secondary 60E05 Keywords: Bessel polynomials, Student-t distribution, linearization coefficients 1 Introduction In this paper we consider the Bessel polynomials qn of degree n n X (n) k qn (u) = αk u , (1) k=0 where n k k (n) (k) 2 n! (2n − k)! 2 αk = 2n k! = . (2) ( k ) (2n)! (n − k)! k! The first examples of these polynomials are u2 q (u) = 1, q (u) = 1 + u, q (u) = 1 + u + . 0 1 2 3 They are normalized according to qn (0) = 1, and thus differ from the polynomials θn (u) in Grosswald’s monograph [12] by (2n)! the constant factor n!2n , i.e. (2n)! θ (u) = q (u). n n!2n n 1 The polynomials θn are sometimes called the reverse Bessel polynomials and n yn(u) = u θn(1/u) the ordinary Bessel polynomials. Two-parameter extensions of these polynomials are studied in [12], and we refer to this work concerning references to the vast literature and the history about Bessel polynomials.