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11: 13.14 Definitions and Basic Properties

… ►The series ►

13.14.6 $$M_{\kappa ,\mu }\left(z\right)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}+\mu }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s}{{\left(1+2\mu \right)}_{s}s!}{z}^s=z^{\frac{1}{2}+\mu }\sum _{n=0}^{\mathrm{\infty }}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\frac{1}{2}+\mu -\kappa }{1+2\mu };2)\frac{{\left(-\frac{1}{2}z\right)}^n}{n!},$$ $2\mu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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13.14.9 $$W_{\kappa ,\pm \frac{1}{2}n}\left(z\right)={(-1)}^{\kappa -\frac{1}{2}n-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}n+\frac{1}{2}}\sum _{k=0}^{\kappa -\frac{1}{2}n-\frac{1}{2}}\left(\genfrac{}{}{0.0pt}{}{\kappa -\frac{1}{2}n-\frac{1}{2}}{k}\right){\left(n+1+k\right)}_{\kappa -k-\frac{1}{2}n-\frac{1}{2}}{(-z)}^k,$$ $\kappa -\frac{1}{2}n-\frac{1}{2}=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.14(i)

Permalink:

http://dlmf.nist.gov/13.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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12: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

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13: 16.14 Partial Differential Equations

… ►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …

14: 19.15 Advantages of Symmetry

… ►Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). … ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). …

15: 16.4 Argument Unity

… ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ 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. … ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. …

16: 20.11 Generalizations and Analogs

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§20.11(i) Gauss Sum

►For relatively prime integers $m,n$ with $n>0$ and $mn$ even, the Gauss sum $G(m,n)$ is defined by … ►

§20.11(ii) Ramanujan’s Theta Function and $q$-Series

… ►Similar identities can be constructed for ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};1;k^2)$, ${}_2{}^{}F_1^{}(\frac{1}{4},\frac{3}{4};1;k^2)$, and ${}_2{}^{}F_1^{}(\frac{1}{6},\frac{5}{6};1;k^2)$. …

17: 16.11 Asymptotic Expansions

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16.11.6 $$\left(\prod _{\mathrm{\ell }=1}^{q+1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=H_{q+1,q}(-z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$;

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $H_{p,q}(z)$: formal infinite series

Permalink:

http://dlmf.nist.gov/16.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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16.11.7 $$\left(\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_q{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_q}{b_1,\mathrm{\dots },b_q};z)\sim H_{q,q}(z{\mathrm{e}}^{\mp \pi \mathrm{i}})+E_{q,q}(z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $H_{p,q}(z)$: formal infinite series

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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16.11.8 $$\left(\prod _{\mathrm{\ell }=1}^{q-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q-1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q-1}}{b_1,\mathrm{\dots },b_q};-z)\sim H_{q-1,q}(z)+E_{q-1,q}(z{e}^{-\pi \mathrm{i}})+E_{q-1,q}(z{e}^{\pi \mathrm{i}}),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series, $H_{p,q}(z)$: formal infinite series and $e_{k,m}$

Permalink:

http://dlmf.nist.gov/16.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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16.11.9 $$\left(\prod _{\mathrm{\ell }=1}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-z)\sim E_{p,q}(z{e}^{-\pi \mathrm{i}})+E_{p,q}(z{e}^{\pi \mathrm{i}}),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $e_{k,m}$

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

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18: 35.10 Methods of Computation

… ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

19: Bibliography R

… ►

Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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External Links:

ISSN 0098-3500, Document

Referenced by:

1st item.

Encodings:

BibTeX

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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

… ►

Elliptic and modular functions from Gauss to Dedekind to Hecke. Cambridge University Press, Cambridge.

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External Links:

ISBN 978-1-107-15938-9, Document, Link, MathReview (Paul M. Jenkins), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:

McGraw-Hill Series in Higher Mathematics

External Links:

MathReview (F. Smithies), ZentralBlatt

Referenced by:

§1.18(x), §2.6(i).

Encodings:

BibTeX

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Principles of Mathematical Analysis. 3rd edition, McGraw-Hill Book Co., New York.

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Notes:

International Series in Pure and Applied Mathematics

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.4(iii).

Encodings:

BibTeX

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20: 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)). … … ►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product … ►