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1: 17.1 Special Notation

§17.1 Special Notation

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$k,j,m,n,r,s$	nonnegative integers.
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►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. ►Another function notation used is the “idem” function: …

2: 17.17 Physical Applications

§17.17 Physical Applications

… ►They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. See Kassel (1995). … ►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

3: 17 q-Hypergeometric and Related Functions

Chapter 17 $q$-Hypergeometric and Related Functions

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4: 23.15 Definitions

§23.15 Definitions

… ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …(Some references refer to $2\mathrm{\ell }$ as the level). …If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. … ►

5: 17.18 Methods of Computation

§17.18 Methods of Computation

►For computation of the $q$-exponential function see Gabutti and Allasia (2008). ►The two main methods for computing basic hypergeometric functions are: (1) numerical summation of the defining series given in §§17.4(i) and 17.4(ii); (2) modular transformations. Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. …

6: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of $a$, $b$, and $x$, and also to complex values. … ►

§8.17(ii) Hypergeometric Representations

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§8.17(iv) Recurrence Relations

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§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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7: George E. Andrews

… ►Andrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. …Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. … ►

17 q-Hypergeometric and Related Functions

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8: 17.4 Basic Hypergeometric Functions

… ►The series (17.4.1) is said to be balanced or Saalschützian when it terminates, $r=s$, $z=q$, and … ►The series (17.4.1) is said to be k-balanced when $r=s$ and … ►The series (17.4.1) is said to be well-poised when $r=s$ and … ►The series (17.4.1) is said to be very-well-poised when $r=s$, (17.4.11) is satisfied, and … ►The series (17.4.1) is said to be nearly-poised when $r=s$ and …

9: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

Encodings:

BibTeX

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Proc. Roy. Soc. London Ser. A 361, pp. 265–275.

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External Links:

FullText, MathReview (A. D. Brjuno), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

Encodings:

BibTeX

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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10: 17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

§17.8 Special Cases of ${}_r{}^{}\psi _r^{}$ Functions

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Quintuple Product Identity

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Bailey’s Bilateral Summations

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Sum Related to (17.6.4)

… ►For similar formulas see Verma and Jain (1983).