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
• Macdonald theory
  • 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 1987 manuscript)
  • 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
  • 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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