# §16.16 Transformations of Variables

## §16.16(i) Reduction Formulas

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

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

see Burchnall and Chaundy (1940, 1941).

## §16.16(ii) Other Transformations

 16.16.8 $\mathop{{F_{1}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma;x,y% \right)=(1-x)^{-\beta}(1-y)^{-\beta^{\prime}}\mathop{{F_{1}}\/}\nolimits\!% \left(\gamma-\alpha;\beta,\beta^{\prime};\gamma;\frac{x}{x-1},\frac{y}{y-1}% \right)=(1-x)^{-\alpha}\mathop{{F_{1}}\/}\nolimits\!\left(\alpha;\gamma-\beta-% \beta^{\prime},\beta^{\prime};\gamma;\frac{x}{x-1},\frac{y-x}{1-x}\right),$ Symbols: $\mathop{{F_{1}}\/}\nolimits\!\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$: first Appell function Permalink: http://dlmf.nist.gov/16.16.E8 Encodings: TeX, pMML, png See also: info for 16.16(ii)
 16.16.9 $\mathop{{F_{2}}\/}\nolimits\!\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{% \prime};x,y\right)=(1-x)^{-\alpha}\mathop{{F_{2}}\/}\nolimits\!\left(\alpha;% \gamma-\beta,\beta^{\prime};\gamma,\gamma^{\prime};\frac{x}{x-1},\frac{y}{1-x}% \right),$
 16.16.10 $\mathop{{F_{4}}\/}\nolimits\!\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y% \right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma^{\prime}\right)\mathop{% \Gamma\/}\nolimits\!\left(\beta-\alpha\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\gamma^{\prime}-\alpha\right)\mathop{\Gamma\/}\nolimits\!\left(\beta% \right)}(-y)^{-\alpha}\mathop{{F_{4}}\/}\nolimits\!\left(\alpha,\alpha-\gamma^% {\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}\right)+\frac{\mathop{% \Gamma\/}\nolimits\!\left(\gamma^{\prime}\right)\mathop{\Gamma\/}\nolimits\!% \left(\alpha-\beta\right)}{\mathop{\Gamma\/}\nolimits\!\left(\gamma^{\prime}-% \beta\right)\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}(-y)^{-\beta}% \mathop{{F_{4}}\/}\nolimits\!\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-% \alpha+1;\frac{x}{y},\frac{1}{y}\right).$

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