expansions in series of hypergeometric functions

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21: 13.2 Definitions and Basic Properties

… ►In effect, the regular singularities of the hypergeometric differential equation at

b

and

\infty

coalesce into an irregular singularity at

\infty

. … ►The first two standard solutions are: … ►The series (13.2.2) and (13.2.3) converge for all

z\in\mathbb{C}

M\left(a,b,z\right)

is entire in

z

and

a

, and is a meromorphic function of

b

. … ►

Kummer’s Transformations

…

22: 15.19 Methods of Computation

§15.19 Methods of Computation

… ►For

z\in\mathbb{R}

it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval

[0,\frac{1}{2}]

. … ►The representation (15.6.1) can be used to compute the hypergeometric function in the sector

|\operatorname{ph}\left(1-z\right)|<\pi

. … ► … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …

23: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►The generalized hypergeometric function

{{}_{p}F_{q}}

with matrix argument

\mathbf{T}\in\boldsymbol{\mathcal{S}}

, numerator parameters

a_{1},\dots,a_{p}

, and denominator parameters

b_{1},\dots,b_{q}

is … ►If

-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}

for some

j, k

satisfying

1\leq j\leq p

1\leq k\leq m

, then the series expansion (35.8.1) terminates. … ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions

{{}_{p}F_{q}}

and

{{}_{p+1}F_{p}}

of matrix argument. A similar result for the

{{}_{0}F_{1}}

function of matrix argument is given in Faraut and Korányi (1994, p. 346). …

24: Bibliography F

… ►

S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii), §15.12(iii).
Encodings:: BibTeX

… ►

J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

ⓘ

External Links:: FullText, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.10.
Encodings:: BibTeX

… ►

W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

ⓘ

Notes:: Reprint with corrections of two books published originally in 1916 and 1936.
External Links:: MathReview
Referenced by:: §2.10(iii).
Encodings:: BibTeX

… ►

C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.

ⓘ

External Links:: ISSN 0030-8730, MathReview (Paolo M. Soardi), ZentralBlatt
Referenced by:: §1.8(i).
Encodings:: BibTeX

… ►

T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §20.7(ii), §20.7(ii).
Encodings:: BibTeX

…

25: 13.31 Approximations

§13.31 Approximations

►

§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of

M\left(a,b,x\right)

and

U\left(a,b,x\right)

that include the intervals

0\leq x\leq\alpha

and

\alpha\leq x<\infty

, respectively, where

\alpha

is an arbitrary positive constant. … ►In Luke (1977a) the following rational approximation is given, together with its rate of convergence. … ►

13.31.1

A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left% (a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s% }}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),

ⓘ

Defines:: $A_{n}(z)$ : expansion (locally)
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.31.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.31(iii), §13.31 and Ch.13

…

26: 16.4 Argument Unity

… ►The function

{{}_{3}F_{2}}\left(a,b,c;d,e;1\right)

is analytic in the parameters

a, b, c, d, e

when its series expansion converges and the bottom parameters are not negative integers or zero. …

27: Bibliography G

… ►

F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

ⓘ

External Links:: ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt
Referenced by:: §17.7(iii).
Encodings:: BibTeX

… ►

G. Gasper and M. Rahman (1990) Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge.

ⓘ

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).
External Links:: ISBN 0-521-35049-2, MathReview (Shaun Cooper), ZentralBlatt
Referenced by:: (17.9.3), Erratum (V1.0.23) for Equation (17.9.3).
Encodings:: BibTeX

… ►

L. Gatteschi (1990) New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §13.9(i), §13.9(ii).
Encodings:: BibTeX

… ►

W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §13.29(iii), §15.19(iii).
Encodings:: BibTeX

… ►

R. A. Gustafson (1987) Multilateral summation theorems for ordinary and basic hypergeometric series in

{\rm U}(n)

. SIAM J. Math. Anal. 18 (6), pp. 1576–1596.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview, ZentralBlatt
Referenced by:: §17.15.
Encodings:: BibTeX

…

28: 13.14 Definitions and Basic Properties

… ►The series … ►In general

M_{\kappa,\mu}\left(z\right)

and

W_{\kappa,\mu}\left(z\right)

are many-valued functions of

z

with branch points at

z=0

and

z=\infty

. The principal branches correspond to the principal branches of the functions

z^{\frac{1}{2}+\mu}

and

U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)

on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i). … ►Although

M_{\kappa,\mu}\left(z\right)

does not exist when

2\mu=-1,-2,-3,\dots

, many formulas containing

M_{\kappa,\mu}\left(z\right)

continue to apply in their limiting form. … ►Except when

z=0

, each branch of the functions

\ifrac{M_{\kappa,\mu}\left(z\right)}{\Gamma\left(2\mu+1\right)}

and

W_{\kappa,\mu}\left(z\right)

is entire in

\kappa

and

\mu

. …

29: Bibliography V

… ►

H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §19.12, §19.5.
Encodings:: BibTeX

… ►

R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

ⓘ

External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

… ►

H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt
Referenced by:: §16.11(ii), §16.11(ii).
Encodings:: BibTeX

… ►

H. Volkmer (2021) Fourier series representation of Ferrers function

{\sf P}

ⓘ

External Links:: FullText
Referenced by:: (14.13.1).
Encodings:: BibTeX

►

H. Volkmer (2023) Asymptotic expansion of the generalized hypergeometric function

{}_{p}F_{q}(z)

z\to\infty

for

p<q

. Anal. Appl. (Singap.) 21 (2), pp. 535–545.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).
Encodings:: BibTeX

…

30: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ►where

{{}_{2}F_{1}}

is the Gauss hypergeometric function (§§15.1 and 15.2(i)). … … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). …