# §16.16 Transformations of Variables

## §16.16(i) Reduction Formulas

 16.16.1 $\displaystyle{F_{1}}\left(\alpha;\beta,\beta^{\prime};\beta+\beta^{\prime};x,y\right)$ $\displaystyle=(1-y)^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\beta+\beta^% {\prime}};\frac{x-y}{1-y}\right),$ 16.16.2 $\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\beta^{\prime};x,% y\right)$ $\displaystyle=(1-y)^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};% \frac{x}{1-y}\right),$ 16.16.3 $\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\alpha;x,y\right)$ $\displaystyle=(1-y)^{-\beta^{\prime}}{F_{1}}\left(\beta;\alpha-\beta^{\prime},% \beta^{\prime};\gamma;x,\frac{x}{1-y}\right),$ 16.16.4 $\displaystyle{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$ $\displaystyle=(1-y)^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};% \gamma;x,\frac{y}{y-1}\right),$ 16.16.5 $\displaystyle{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;\gamma;x,y\right)$ $\displaystyle=(1-y)^{\alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,\beta\atop% \gamma};x+y-xy\right),$ 16.16.6 $\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\alpha+\beta-\gamma+1;x(1-y),y(1% -x)\right)$ $\displaystyle={{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x\right){{}_{2}F_{1% }}\left({\alpha,\beta\atop\alpha+\beta-\gamma+1};y\right).$

See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. An extension of (16.16.6) is given by

 16.16.7 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\sum_{k=% 0}^{\infty}\frac{{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}{\left(% \alpha+\beta-\gamma-\gamma^{\prime}+1\right)_{k}}}{{\left(\gamma\right)_{k}}{% \left(\gamma^{\prime}\right)_{k}}k!}x^{k}y^{k}{{}_{2}F_{1}}\left({\alpha+k,% \beta+k\atop\gamma+k};x\right){{}_{2}F_{1}}\left({\alpha+k,\beta+k\atop\gamma^% {\prime}+k};y\right);$

see Burchnall and Chaundy (1940, 1941).

## §16.16(ii) Other Transformations

 16.16.8 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-x)^{-\beta}(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\gamma-\alpha;\beta,\beta^{\prime};\gamma;\frac% {x}{x-1},\frac{y}{y-1}\right)=(1-x)^{-\alpha}{F_{1}}\left(\alpha;\gamma-\beta-% \beta^{\prime},\beta^{\prime};\gamma;\frac{x}{x-1},\frac{y-x}{1-x}\right),$ ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$: first Appell function Permalink: http://dlmf.nist.gov/16.16.E8 Encodings: TeX, pMML, png See also: Annotations for §16.16(ii), §16.16 and Ch.16
 16.16.9 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=(1-% x)^{-\alpha}{F_{2}}\left(\alpha;\gamma-\beta,\beta^{\prime};\gamma,\gamma^{% \prime};\frac{x}{x-1},\frac{y}{1-x}\right),$ ⓘ Symbols: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$: second Appell function Permalink: http://dlmf.nist.gov/16.16.E9 Encodings: TeX, pMML, png See also: Annotations for §16.16(ii), §16.16 and Ch.16
 16.16.10 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\frac{\Gamma\left(% \gamma^{\prime}\right)\Gamma\left(\beta-\alpha\right)}{\Gamma\left(\gamma^{% \prime}-\alpha\right)\Gamma\left(\beta\right)}(-y)^{-\alpha}{F_{4}}\left(% \alpha,\alpha-\gamma^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\Gamma\left(\gamma^{\prime}\right)\Gamma\left(\alpha-\beta\right% )}{\Gamma\left(\gamma^{\prime}-\beta\right)\Gamma\left(\alpha\right)}(-y)^{-% \beta}{F_{4}}\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-\alpha+1;\frac{x% }{y},\frac{1}{y}\right).$ ⓘ Symbols: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$: fourth Appell function and $\Gamma\left(\NVar{z}\right)$: gamma function Permalink: http://dlmf.nist.gov/16.16.E10 Encodings: TeX, pMML, png See also: Annotations for §16.16(ii), §16.16 and Ch.16

For quadratic transformations of Appell functions see Carlson (1976).