§16.6 Transformations of Variable

Notes:: For (16.6.1) see Whipple (1927). For (16.6.2) see Bailey (1929).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.6

¶ Quadratic

Keywords:: generalized hypergeometric functions

16.6.1 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,b,c\atop a-b+1,a-c+1};z\right)=(1-z)^{{-a}}\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)\atop a-b+1,a-c+1};\frac{-4z}{(1-z)^{2}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : 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

¶ Cubic

Keywords:: generalized hypergeometric functions

16.6.2 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)=(1-z)^{{-a}}\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : 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

For Kummer-type transformations of $\mathop{{{}_{{2}}F_{{2}}}\/}\nolimits$ functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5).