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

ⓘ

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

ⓘ

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, : 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

ⓘ

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, : 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).

16.5 Integral Representations and Integrals 16.7 Relations to Other Functions

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov