Special functions

• Societies and conference series
  • SIAM Activity Group on Orthogonal Polynomials and Special Functions
  • OPSFA conference series (Orthogonal Polynomials, Special Functions, and Applications)
  • Society for Special Functions & their Applications (SSFA), India
  • Foundations of Computational Mathematics (FoCM)
• Handbooks
  • Bateman project
    • Higher Transcendental Functions, Vols. 1,2,3
    • Tables of Integral Transforms, Vols. 1,2
    • Askey-Bateman project
  • Digital Library of Mathematical Functions (DLMF)
    • DLMF online
    • Handbook of Mathematical Functions, Abramowitz and Stegun (scan provided by
      Alan P. Sexton, University of Birmingham)
    • DRMF (NIST Digital Repository of Mathematical Formulae)
  • Askey scheme
    • Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials by R. Askey and J. Wilson, Mem. Amer. Math. Soc. 54 (1985), no. 319.
    • Hypergeometric orthogonal polynomials and their q-analogues by Roelof Koekoek,
      Peter A. Lesky and René Swarttouw; Springer-Verlag, 2010.
    • Chapters 9 (Hypergeometric orthogonal polynomials) and 14 (Basic hypergeometric polynomials) of the above book are a slightly extended and updated version of
      The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue by
      R. Koekoek & R.F. Swarttouw, Report no. 98-17, 1998, Delft Technical University.
    • Online version of the above report
    • Askey scheme and q-Askey scheme charts in various formats
    • Askey scheme with pictures (jpg file)
    • The origin of the Askey scheme:
      • Michael Hoare's seven-fold way of orthogonal polynomials and seven-fold way of probability distributions as presented by him at an Oberwalfach meeting in 1977 on Combinatorics and Special Functions.
      • Dick Askey's account of Michael Hoare's presentation and its aftermath (in pp. v and vi of Tom Koornwinder's Foreword to the book Hypergeometric orthogonal polynomials and their q-analogues by R. Koekoek, P.A. Lesky and R.F. Swarttouw, Springer-Verlag, 2010.
      • J. Labelle, Tableau d'Askey, in: Orthogonal polynomials and applications (Bar-le-Duc, 1984), Lecture Notes in Math. 1171, pp. xxxvi--xxxvii, Springer-Verlag, 1985 (Labelle's poster with short introduction).
      • J. Labelle, Askey's scheme of hypergeometric orthogonal polynomials, 1990 (Labelle's poster, with details spread over a number of pages).
    • Josef Meixner: His life and his orthogonal polynomials, paper by Paul Butzer and Tom Koornwinder, Indag. Math. (N.S.) 30 (2019), 250-264; arXiv:1609.02588v3 [math.HO].
    • Historical page in Ukrainian about Krawtchouk
  • Gradshteyn and Ryzhik, Table of Integrals, Series, and Products
    • Home Page
    • Eighth edition at Elsevier
    • Daniel Zwillinger's errata page
    • Proofs of some of the formulas (by Victor Hugo Moll)
  • Handbook of Continued Fractions for Special Functions (A.A.M. Cuyt, V. Petersen,
    B. Verdonk, H. Waadeland, W.B. Jones)
    • book at Springer-Verlag (2008)
    • online software
• Ian G. Macdonald passed away on August 8, 2023, at the age of 94. Among others, he introduced Macdonald polynomials:
  • A new class of symmetric functions by I.G. Macdonald, Séminaire Lotharingien de Combinatoire, B20a (1988), 41 pp.
  • Symmetric functions and Hall polynomials by I.G. Macdonald, Oxford University Press,
    Second ed., 1995.
  • Orthogonal polynomials associated with root systems by I.G. Macdonald, Séminaire Lotharingien Combinatoire 45 (2000), B45a, 40 pp. (from his 1988 manuscript, which originated from his 1987 manuscripts "Jacobi polynomials I" and
    “Jacobi polynomials II (unequal labels)”)
  • Affine Hecke algebras and orthogonal polynomials by I.G. Macdonald, Séminaire Bourbaki 37 (1994-1995), exp. no. 797 (1995), 189-207.
  • Affine Hecke algebras and orthogonal polynomials by I.G. Macdonald, Cambridge University Press, 2003.
  • Hypergeometric functions I and Hypergeometric functions II (q-analogues) by I.G. Macdonald, arXiv:1309.4568 and arXiv:1309.5208 (from his manuscripts in 1987 or 1988)
  • I.G. Macdonald's honorary doctorate at UvA, January 2002.
  • Macdonald polynomials web page (by Mike Zabrocki)
  • The symmetric functions catalog (by Per Alexandersson).
  • Generalized Kostka polynomials: 2005 workshop at AIM and introductory webpage
• Dick Askey
  • Richard Allen "Dick" Askey, June 4, 1933 - October 9, 2019
  • Richard Askey day on 15 June 2022 (part of OPSFA 16)
    • Videos and presentations of invited speakers
  • Travel diaries by Liz Askey, September 1987 - January 1988 (edited by Tom Koornwinder):
    • trip to U.S.S.R, September 1987
    • trip to Japan, October-November 1987
    • trip to Australia, November-December 1987
    • trip to India, December 1987 - January 1988
  • See also the slides of my lecture on 15 June 2022 about the first three diaries.
  • a picture gallery at Dick Askey's eightieth birthday.
• Bibliographies
  • Bibliography of Elliptic Hypergeometric Functions (by Hjalmar Rosengren)
  • Gabor Szegö home page (by Paul Nevai)
  • Krawtchouk Polynomials Home Page
• Wolfram
  • The Wolfram Functions Site
  • The Integrator
• Further online tools
  • Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)
  • OPQ: a Matlab suite of programs for generating orthogonal polynomials and related quadrature rules by Walter Gautschi
  • Inverse Symbolic Calculator (Centre for Experimental and Constructive Mathematics)
  • Numbers, constants and computation (by Xavier Gourdon and Pascal Sebah)
  • DDMF (Dynamical Dictionary of Mathematical Functions)
  • Fungrim, an open source library of formulas and data for special functions by Fredrik Johansson. See further information.
• Frozen projects
  • The Heun Project
  • Chromatic Derivatives and Chromatic Approximations Home Page (Aleksandar Ignjatovic)

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