generalized hypergeometric series

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1: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ►Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in

z

. … ►If none of the

a_{j}

is a nonpositive integer, then the radius of convergence of the series (16.2.1) is

1

, and outside the open disk

|z|<1

the generalized hypergeometric function is defined by analytic continuation with respect to

z

. … ►However, when one or more of the top parameters

a_{j}

is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in

z

. … ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …

2: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

3: 16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

§16.10 Expansions in Series of ${{}_{p}F_{q}}$ Functions

… ►Expansions of the form

\sum_{n=1}^{\infty}(\pm 1)^{n}{{}_{p}F_{p+1}}\left(\mathbf{a};\mathbf{b};-n^{2% }z^{2}\right)

are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

4: 34.4 Definition: $\mathit{6j}$ Symbol

… ►For alternative expressions for the

\mathit{6j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{4}F_{3}}

of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

5: 34.2 Definition: $\mathit{3j}$ Symbol

… ►For alternative expressions for the

\mathit{3j}

symbol, written either as a finite sum or as other terminating generalized hypergeometric series

{{}_{3}F_{2}}

of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

6: Bibliography B

… ►

W. N. Bailey (1928) Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

►

W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

W. N. Bailey (1964) Generalized Hypergeometric Series. Stechert-Hafner, Inc., New York.

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External Links:: MathReview
Referenced by:: §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §16.4(iii), §17.1, §17.4(i), Ch.17.
Encodings:: BibTeX

…

7: Bibliography P

… ►

W. F. Perger, A. Bhalla, and M. Nardin (1993) 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.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

…

8: Bibliography W

… ►

F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

…

9: 16.4 Argument Unity

… ► … ►

§16.4(v) Bilateral Series

… ►

10: 16.11 Asymptotic Expansions

… ►

§16.11(i) Formal Series

… ►

16.11.6

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q+1}F_{q}}% \left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=H_{q+1,q}(-z),

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

;

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : 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

… ►

16.11.7

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q}F_{q}}% \left({a_{1},\dots,a_{q}\atop b_{1},\dots,b_{q}};z\right)\sim H_{q,q}(z{% \mathrm{e}}^{\mp\pi\mathrm{i}})+E_{q,q}(z).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\sim$ : 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, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : 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

… ►

16.11.8

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1}\Gamma\left(a_{\ell}\right)% }{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{q-1}F_{q}}% \left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{q}};-z\right)\sim H_{q-1,q}(z)+% E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{\pi\mathrm{i}}),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\sim$ : 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, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : 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

… ►

16.11.9

\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}\Gamma\left(a_{\ell}\right)}{% \prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}\right)}}\right){{}_{p}F_{q}}% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-z\right)\sim E_{p,q}(ze^{-% \pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ 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, $\sim$ : 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, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : 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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