relation to generalized hypergeometric functions

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41: Bibliography M

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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

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§13.12.

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Euler-type transformations for the generalized hypergeometric function ${}_{r+2}{}^{}F_{r+1}^{}(x)$ . Z. Angew. Math. Phys. 62 (1), pp. 31–45.

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ISSN 0044-2275, Document, FullText, MathReview (Mridula Garg), ZentralBlatt

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§16.6, §16.6.

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A class of generalized hypergeometric summations. J. Comput. Appl. Math. 87 (1), pp. 79–85.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§16.10.

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On a Kummer-type transformation for the generalized hypergeometric function ${}_2{}^{}F_2^{}$ . J. Comput. Appl. Math. 157 (2), pp. 507–509.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

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§16.6.

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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

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§17.15.

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42: Bibliography B

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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1st item.

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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MathReview (C. J. Bouwkamp)

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2nd item, 1st item.

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Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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Document, MathReview (C. V. Coffman), ZentralBlatt

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§10.21(xi).

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An analytic continuation formula for the generalized hypergeometric function. SIAM J. Math. Anal. 19 (5), pp. 1249–1251.

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ISSN 0036-1410, Document, MathReview (H. M. Srivastava), ZentralBlatt

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§16.8(ii).

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Generalized hypergeometric functions at unit argument. Proc. Amer. Math. Soc. 114 (1), pp. 145–153.

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ISSN 0002-9939, FullText, MathReview (R. K. Raina), ZentralBlatt

Referenced by:

§16.4(ii), §16.8(ii).

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43: 15.4 Special Cases

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§15.4(i) Elementary Functions

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§15.4(ii) Argument Unity

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Chu–Vandermonde Identity

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§15.4(iii) Other Arguments

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44: Richard A. Askey

… ►Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and related areas. …One of his most influential papers Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials (with J. …Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. … ►

5 Gamma Function

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16 Generalized Hypergeometric Functions & Meijer G-Function

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45: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. … … ►

§14.21(iii) Properties

►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …

46: 18.33 Polynomials Orthogonal on the Unit Circle

… ►For an alternative and more detailed approach to the recurrence relations, see §18.33(vi). … ►

Szegő–Askey

… ►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). Askey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. … ►Equivalent to the recurrence relations (18.33.23), (18.33.24) are the inverse Szegő recurrence relations …

47: Bibliography K

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Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

Referenced by:

§22.9(iv), §22.9(v).

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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

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§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

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§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

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§15.14, §15.14.

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Some special cases of the generalized hypergeometric function ${}_{q+1}{}^{}F_q^{}$ . J. Comput. Appl. Math. 78 (1), pp. 79–95.

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ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

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§16.7.

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48: 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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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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MathReview (H. O. Pollak), ZentralBlatt

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§5.6(i), §6.8, §7.8, §8.10.

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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MathReview Entry, ZentralBlatt

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§3.8(vi).

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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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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

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49: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

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§31.10(i).

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Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.

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ISBN 981-02-4353-7, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§14.29.

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A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt

Referenced by:

§16.11(ii), §16.11(ii).

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

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§29.8, §29.8.

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Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

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§16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

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50: 33.14 Definitions and Basic Properties

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

… ►where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and … ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

►For nonzero values of $ϵ$ and $r$ the function $h(ϵ,\mathrm{\ell };r)$ is defined by … ►

§33.14(v) Wronskians

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