Special functions

Societies and conference series

Handbooks
 Bateman project

Digital Library of Mathematical Functions (DLMF)

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 qanalogues
by Roelof Koekoek,
Peter A. Lesky and René Swarttouw;
SpringerVerlag, 2010.

Chapters 9 (Hypergeometric orthogonal polynomials) and 14
(Basic hypergeometric polynomials) of the above
book are a slightly extended and updated version of
The Askeyscheme
of hypergeometric orthogonal polynomials and its qanalogue
by
R. Koekoek & R.F. Swarttouw,
Report no. 9817, 1998, Delft Technical University.

Online version of the above report

Askey scheme and qAskey scheme charts in various formats

Askey scheme with pictures (jpg file)

The origin of the Askey scheme:

Michael Hoare's sevenfold way of
orthogonal polynomials and
sevenfold 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 qanalogues
by R. Koekoek, P.A. Lesky and R.F. Swarttouw, SpringerVerlag, 2010.

J. Labelle,
Tableau d'Askey,
in: Orthogonal polynomials and applications (BarleDuc, 1984),
Lecture Notes in Math. 1171, pp. xxxvixxxvii, SpringerVerlag, 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), 250264;
arXiv:1609.02588v3 [math.HO].

Historical page in Ukrainian about Krawtchouk

Gradshteyn and Ryzhik,
Table of Integrals, Series, and Products

Handbook of Continued Fractions for Special Functions
(A.A.M. Cuyt, V. Petersen,
B. Verdonk, H. Waadeland, W.B. Jones)

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 (19941995), exp. no. 797 (1995), 189207.

Affine Hecke algebras
and orthogonal polynomials by I.G. Macdonald,
Cambridge University Press, 2003.

Hypergeometric functions I
and Hypergeometric functions II
(qanalogues) 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

Bibliographies

Wolfram

Further online tools

Frozen projects
to Tom Koornwinder's home page