basic hypergeometric functions

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21: Bibliography G

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Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt

Referenced by:

§17.7(iii).

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Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

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Notes:

A revised and updated edition, with three new chapters, was published in 2004 by Cambridge University Press (Encyclopedia of Mathematics and its Applications, vol. 96).

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ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

(17.9.3), Erratum (V1.0.23) for Equation (17.9.3).

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Basic Hypergeometric Series. Second edition, Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge.

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Notes:

With a foreword by Richard Askey

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ISBN 0-521-83357-4, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§17.1, §17.1, §17.10, (17.11.1), (17.11.2), (17.11.3), §17.13, §17.16, (17.4.5), (17.4.6), (17.4.7), (17.4.8), §17.4(i), §17.5, §17.6(i), §17.6(ii), §17.6(iii), §17.6(iv), §17.6(v), §17.7(i), §17.7(ii), §17.7(iii), §17.8, (17.9.3), §17.9(i), §17.9(ii), §17.9(iii), Ch.17, §18.27(i), §18.27(i), §18.27(ii), §18.27(iii), §18.27(iv), (18.28.24), (18.28.25), §18.28(i), §18.28(ii), §18.28(v), §18.28(viii), §26.19, Erratum (V1.0.10) for Section 17.1, Erratum (V1.0.23) for Equation (17.9.3).

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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.29(iii), §15.19(iii).

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Multilateral summation theorems for ordinary and basic hypergeometric series in $\mathrm{U}(n)$ . SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

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ISSN 0036-1410, Document, MathReview, ZentralBlatt

Referenced by:

§17.15.

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22: Bibliography M

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Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

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FullText

Referenced by:

§35.7(ii), §35.8(iii).

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§13.12.

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

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Balanced ${}_3{}^{}\Theta _2^{}$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1), pp. 93–187.

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ISSN 0001-8708, Document, MathReview (Jiang Zeng), ZentralBlatt

Referenced by:

§17.15.

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23: Bibliography F

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Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

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Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.

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Notes:

With a foreword by George E. Andrews

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ISBN 0-8218-1524-5, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.1, §17.16, §17.6(ii), Ch.17, Notations F.

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Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.

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Notes:

Double-precision Fortran

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ISSN 0021-9991, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.19(i), 1st item, 2nd item.

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24: Bibliography K

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On the evaluation of the Gauss hypergeometric function. C. R. Acad. Bulgare Sci. 45 (6), pp. 35–36.

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ISSN 0861-1459, MathReview, ZentralBlatt

Referenced by:

§15.15, §15.19(i).

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On the zeros of some generalized hypergeometric functions. J. Math. Anal. Appl. 243 (2), pp. 249–260.

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ISSN 0022-247X, Document, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§16.9.

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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

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Code, Document, MathReview (G. Gasper), ZentralBlatt

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1st item.

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25: 13.2 Definitions and Basic Properties

§13.2 Definitions and Basic Properties

… ►In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. … ► $M(a,b,z)$ is entire in $z$ and $a$, and is a meromorphic function of $b$. … ►For $U(a,b,z)$ see (13.2.6). … ►

Kummer’s Transformations

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26: 18.38 Mathematical Applications

… ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►

Complex Function Theory

►The Askey–Gasper inequality …For the generalized hypergeometric function ${}_3{}^{}F_2^{}$ see (16.2.1). … ►

Non-Classical Weight Functions

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27: Bibliography P

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Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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ZentralBlatt

Referenced by:

§5.1, Notations F.

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§16.6.

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Exponentially small expansions of the confluent hypergeometric functions. Appl. Math. Sci. (Ruse) 7 (133-136), pp. 6601–6609.

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External Links:

ISSN 1312-885X, Document, MathReview Entry

Referenced by:

§13.7(iii), §13.7(iii).

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

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Document, Code, ZentralBlatt

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1st item.

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A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.

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Notes:

Includes Basic code.

External Links:

Document

Referenced by:

§8.28(iv).

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28: 31.3 Basic Solutions

§31.3 Basic Solutions

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§31.3(i) Fuchs–Frobenius Solutions at $z=0$

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§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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§31.3(iii) Equivalent Expressions

… ►The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).

29: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(i) Definitions and Basic Properties

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§8.17(ii) Hypergeometric Representations

… ►For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). … ►

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

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30: 33.2 Definitions and Basic Properties

§33.2 Definitions and Basic Properties

… ►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by …where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and … ►The functions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ are defined by …where $W_{\kappa ,\mu }\left(z\right)$, $U(a,b,z)$ are defined in §§13.14(i) and 13.2(i), …