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Basic hypergeometric series. 2nd ed. (English) Zbl 1129.33005

Encyclopedia of Mathematics and Its Applications 96. Cambridge: Cambridge University Press (ISBN 0-521-83357-4/hbk). xxvi, 428 p. (2004).
From the text: This revised and expanded new edition will surely continue to meet the needs for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or \(q\)-series. Simplicity, clarity, deductive proofs, thoughtfully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, orthogonal polynomials in several variables, and generating functions.
Chapters 9-11 are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base \(q\)-series.
Some sections and exercises have been added to reflect recent developments, and the Bibliography has been revised to maintain its comprehensiveness.
For the review of the first edition see Zbl 0695.33001.

MSC:

33Dxx Basic hypergeometric functions
33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
33-02 Research exposition (monographs, survey articles) pertaining to special functions
05A30 \(q\)-calculus and related topics
05E35 Orthogonal polynomials (combinatorics) (MSC2000)

Citations:

Zbl 0695.33001
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Digital Library of Mathematical Functions:

§17.10 Transformations of ψ r r Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.11.1) ‣ §17.11 Transformations of q -Appell Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.11.2) ‣ §17.11 Transformations of q -Appell Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.11.3) ‣ §17.11 Transformations of q -Appell Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.13 Integrals ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.16 Mathematical Applications ‣ Applications ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.1 Special Notation ‣ Notation ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.1 Special Notation ‣ Notation ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.4.5) ‣ §17.4(iii) q -Appell Functions ‣ §17.4 Basic Hypergeometric Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.4.6) ‣ §17.4(iii) q -Appell Functions ‣ §17.4 Basic Hypergeometric Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.4.7) ‣ §17.4(iii) q -Appell Functions ‣ §17.4 Basic Hypergeometric Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
(17.4.8) ‣ §17.4(iii) q -Appell Functions ‣ §17.4 Basic Hypergeometric Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.4(i) ϕ s r Functions ‣ §17.4 Basic Hypergeometric Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.5 ϕ 0 0 , ϕ 0 1 , ϕ 1 1 Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.6(iii) Contiguous Relations ‣ §17.6 ϕ 1 2 Function ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.6(ii) ϕ 1 2 Transformations ‣ §17.6 ϕ 1 2 Function ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.6(i) Special Values ‣ §17.6 ϕ 1 2 Function ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.6(iv) Differential Equations ‣ §17.6 ϕ 1 2 Function ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.6(v) Integral Representations ‣ §17.6 ϕ 1 2 Function ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.7(iii) Other ϕ s r Functions ‣ §17.7 Special Cases of Higher ϕ s r Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.7(ii) ϕ 2 3 Functions ‣ §17.7 Special Cases of Higher ϕ s r Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.7(i) ϕ 2 2 Functions ‣ §17.7 Special Cases of Higher ϕ s r Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.8 Special Cases of ψ r r Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.9(iii) Further ϕ s r Functions ‣ §17.9 Further Transformations of ϕ r + r 1 Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.9(ii) → ϕ 2 3 ϕ 2 3 ‣ §17.9 Further Transformations of ϕ r + r 1 Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
§17.9(i) → ϕ 1 2 ϕ 2 2 , ϕ 1 3 , or ϕ 2 3 ‣ §17.9 Further Transformations of ϕ r + r 1 Functions ‣ Properties ‣ Chapter 17 q -Hypergeometric and Related Functions
Chapter 17 q -Hypergeometric and Related Functions
§18.27(iii) Big q -Jacobi Polynomials ‣ §18.27 q -Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.27(ii) q -Hahn Polynomials ‣ §18.27 q -Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.27(i) Introduction ‣ §18.27 q -Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.27(iv) Little q -Jacobi Polynomials ‣ §18.27 q -Hahn Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.28(ii) Askey–Wilson Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.28(i) Introduction ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.28(viii) q -Racah Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§18.28(v) Continuous q -Ultraspherical Polynomials ‣ §18.28 Askey–Wilson Class ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
§26.19 Mathematical Applications ‣ Applications ‣ Chapter 26 Combinatorial Analysis
Erratum (V1.0.10) for Section 17.1 ‣ Other Changes ‣ Version 1.0.10 (August 7, 2015) ‣ Errata