16 Generalized Hypergeometric Functions & Meijer G-Function Generalized Hypergeometric Functions 16.5 Integral Representations and Integrals 16.7 Relations to Other Functions

§16.6 Transformations of Variable

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Keywords:: generalized hypergeometric functions, transformation of variable
Notes:: For (16.6.1) see Whipple (1927). For (16.6.2) see Bailey (1929).
Referenced by:: Erratum (V1.0.10) for References, Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/16.6
Addition (effective with 1.0.10):: References to Choi and Rathie (2013) and Wang and Rathie (2013) were added at the end of this section.
Addition (effective with 1.0.7):: The reference to Miller and Paris (2011) has been added at the end of this section.
See also:: Annotations for Ch.16

Quadratic

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Keywords:: generalized hypergeometric functions, quadratic, transformation of variable
See also:: Annotations for §16.6 and Ch.16

16.6.1		${{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};z\right)=(1-z)^{-a}{{}_{3}F_{2}}% \left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)\atop a-b+1,a-c+1};\frac{-4z}{(1-z% )^{2}}\right).$
ⓘ 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, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Referenced by: §16.6 Permalink: http://dlmf.nist.gov/16.6.E1 Encodings: TeX, pMML, png See also: Annotations for §16.6, §16.6 and Ch.16

Cubic

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Keywords:: Kummer-type transformations, cubic, generalized hypergeometric functions, transformation of variable
See also:: Annotations for §16.6 and Ch.16

16.6.2		${{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).$
ⓘ 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, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Referenced by: §16.6 Permalink: http://dlmf.nist.gov/16.6.E2 Encodings: TeX, pMML, png See also: Annotations for §16.6, §16.6 and Ch.16

For Kummer-type transformations of ${{}_{2}F_{2}}$ functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is available. 16.5 Integral Representations and Integrals 16.7 Relations to Other Functions