generalized hypergeometric series

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21: Bibliography S

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Asymptotic series of generalized Lambert $W$ function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.

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External Links:

FullText, ZentralBlatt

Referenced by:

§4.13, §4.13.

Encodings:

BibTeX

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Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.

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External Links:

ISBN 9780521090612, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

Ch.15, §16.2(iii), §16.4(ii), §16.4(iii), §16.4(v), Ch.16, §17.1, §17.4(i), Ch.17.

Encodings:

BibTeX

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On the logarithmic solutions of the generalized hypergeometric equation when $p=q+1$ . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.

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External Links:

Document, MathReview (H. Bateman), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

Encodings:

BibTeX

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Computation of angular momentum coefficients using sets of generalized hypergeometric functions. Comput. Phys. Comm. 22 (2-3), pp. 297–302.

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External Links:

Document

Referenced by:

§34.13.

Encodings:

BibTeX

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22: 18.26 Wilson Class: Continued

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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

►For the definition of generalized hypergeometric functions see §16.2. … ►

18.26.2 $$S_n(y^2;a,b,c)={\left(a+b\right)}_n{\left(a+c\right)}_n{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,a+iy,a-iy}{a+b,a+c};1).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $y$: real variable and $n$: nonnegative integer

Referenced by:

§18.26(i), §18.26(ii), §18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(i), §18.26 and Ch.18

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§18.26(iv) Generating Functions

►For the hypergeometric function ${}_2{}^{}F_1^{}$ see §§15.1 and 15.2(i). …

23: 7.14 Integrals

… ►In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.

24: 18.23 Hahn Class: Generating Functions

… ►For the definition of generalized hypergeometric functions see §16.2. ►

Hahn

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18.23.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-x}{\alpha +1};-z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{x-N}{\beta +1};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n}{{\left(\beta +1\right)}_{n}n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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18.23.2 $${}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{-x,-x+\beta +N+1}{-};-z){}_2{}^{}F_0^{}(\genfrac{}{}{0pt}{}{x-N,x+\alpha +1}{-};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\alpha +1\right)}_n}{n!}{Q}_n(x;\alpha ,\beta ,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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, $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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18.23.6 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a+\mathrm{i}x}{2\mathrm{\Re }a};-\mathrm{i}z){}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\overline{b}-\mathrm{i}x}{2\mathrm{\Re }b};\mathrm{i}z)=\sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x;a,b,\overline{a},\overline{b})}{{\left(2\mathrm{\Re }a\right)}_n{\left(2\mathrm{\Re }b\right)}_n}{z}^n.$$

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Symbols:

${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.23

Permalink:

http://dlmf.nist.gov/18.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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25: 17.17 Physical Applications

§17.17 Physical Applications

… ►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys. ►Quantum groups also apply $q$-series extensively. …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. …

26: 8.27 Approximations

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Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for $\mathrm{\Gamma }(a,\omega z)$ (by specifying parameters) with , and $\gamma (a,\omega z)$ with $0\le \omega \le 1$; see also Temme (1994b, §3).

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Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$-plane that exclude $z=0$ and are valid for .

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§8.27(ii) Generalized Exponential Integral

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Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

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27: 2.10 Sums and Sequences

… ►The asymptotic behavior of entire functions defined by Maclaurin series can be approached by converting the sum into a contour integral by use of the residue theorem and applying the methods of §§2.4 and 2.5. … ►

2.10.19 $${}_0{}^{}F_2^{}(-;1,1;x)=\sum _{j=0}^{\mathrm{\infty }}\frac{x^j}{{(j!)}^3}.$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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 and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/2.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(iii), §2.10(iii), §2.10 and Ch.2

… ►Hence … ►For generalizations and other examples see Olver (1997b, Chapter 8), Ford (1960), and Berndt and Evans (1984). … ►What is the asymptotic behavior of $f_n$ as $n\to \mathrm{\infty }$ or $n\to -\mathrm{\infty }$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …

28: Bibliography L

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Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

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External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

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New series expansions for the confluent hypergeometric function $M(a,b,z)$ . Appl. Math. Comput. 235, pp. 26–31.

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External Links:

ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.11, §13.11.

Encodings:

BibTeX

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New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.

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External Links:

ISSN 1019-7168, Document, FullText, MathReview (Jochen Denzler), ZentralBlatt

Referenced by:

§15.19(i), §15.19(i).

Encodings:

BibTeX

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Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.24(ii).

Encodings:

BibTeX

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29: 33.23 Methods of Computation

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§33.23(i) Methods for the Confluent Hypergeometric Functions

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§33.23(ii) Series Solutions

… ►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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Convergence Properties

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§35.8(iv) General Properties

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Confluence

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