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DLMF: §16.16 Transformations of Variables ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function

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16 Generalized Hypergeometric Functions & Meijer G-Function Two-Variable Hypergeometric Functions 16.15 Integral Representations and Integrals 16.17 Definition

§16.16 Transformations of Variables

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Keywords:: Appell functions, transformations of variables
Permalink:: http://dlmf.nist.gov/16.16
See also:: Annotations for Ch.16

Contents

§16.16(i) Reduction Formulas

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Keywords:: Appell functions, reduction formulas, relations to hypergeometric functions, transformations of variables
Permalink:: http://dlmf.nist.gov/16.16.i
Addition (effective with 1.1.12):: Equation (16.16.5_5) was added.
See also:: Annotations for §16.16 and Ch.16

16.16.1		$\displaystyle{F_{1}}\left(\alpha;\beta,\beta^{\prime};\beta+\beta^{\prime};x,y\right)$	$\displaystyle=(1-y)^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\beta+\beta^% {\prime}};\frac{x-y}{1-y}\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function Permalink: http://dlmf.nist.gov/16.16.E1 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16
16.16.2		$\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\beta^{\prime};x,% y\right)$	$\displaystyle=(1-y)^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};% \frac{x}{1-y}\right),$
ⓘ Symbols: ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function Permalink: http://dlmf.nist.gov/16.16.E2 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16
16.16.3		$\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\alpha;x,y\right)$	$\displaystyle=(1-y)^{-\beta^{\prime}}{F_{1}}\left(\beta;\alpha-\beta^{\prime},% \beta^{\prime};\gamma;x,\frac{x}{1-y}\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma},% \NVar{\gamma^{\prime}};\NVar{x},\NVar{y}\right)$ : second Appell function Permalink: http://dlmf.nist.gov/16.16.E3 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16
16.16.4		$\displaystyle{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$	$\displaystyle=(1-y)^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};% \gamma;x,\frac{y}{y-1}\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function Permalink: http://dlmf.nist.gov/16.16.E4 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16
16.16.5		$\displaystyle{F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;\gamma;x,y\right)$	$\displaystyle=(1-y)^{\alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,\beta\atop% \gamma};x+y-xy\right),$
ⓘ Symbols: ${F_{3}}\left(\NVar{\alpha},\NVar{\alpha^{\prime}};\NVar{\beta},\NVar{\beta^{% \prime}};\NVar{\gamma};\NVar{x},\NVar{y}\right)$ : third Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function Permalink: http://dlmf.nist.gov/16.16.E5 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16
16.16.5_5		$\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\beta;x(1-y),y(1-x)\right)$	$\displaystyle=(1-x)^{-\alpha}(1-y)^{-\alpha}{F_{1}}\left(\alpha;\gamma-\beta,% \alpha-\gamma+1;\gamma;\frac{x}{x-1},\frac{xy}{(1-x)(1-y)}\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function Referenced by: §16.16(i), Erratum (V1.1.12) for Additions Permalink: http://dlmf.nist.gov/16.16.E5_5 Encodings: TeX, pMML, png Addition (effective with 1.1.12): This equation was added. See also: Annotations for §16.16(i), §16.16 and Ch.16
16.16.6		$\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\alpha+\beta-\gamma+1;x(1-y),y(1% -x)\right)$	$\displaystyle={{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x\right){{}_{2}F_{1% }}\left({\alpha,\beta\atop\alpha+\beta-\gamma+1};y\right).$
ⓘ Symbols: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function and ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function Referenced by: §16.16(i) Permalink: http://dlmf.nist.gov/16.16.E6 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16

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		${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\sum_{k=% 0}^{\infty}\frac{{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}{\left(% \alpha+\beta-\gamma-\gamma^{\prime}+1\right)_{k}}}{{\left(\gamma\right)_{k}}{% \left(\gamma^{\prime}\right)_{k}}k!}x^{k}y^{k}{{}_{2}F_{1}}\left({\alpha+k,% \beta+k\atop\gamma+k};x\right){{}_{2}F_{1}}\left({\alpha+k,\beta+k\atop\gamma^% {\prime}+k};y\right);$
ⓘ Symbols: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial) and $!$ : factorial (as in $n!$ ) Permalink: http://dlmf.nist.gov/16.16.E7 Encodings: TeX, pMML, png See also: Annotations for §16.16(i), §16.16 and Ch.16

see Burchnall and Chaundy (1940, 1941).

§16.16(ii) Other Transformations

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Keywords:: Appell functions, generalized hypergeometric functions, quadratic, transformations of variables, with two variables
Notes:: See Erdélyi et al. (1953a, §5.11).
Permalink:: http://dlmf.nist.gov/16.16.ii
See also:: Annotations for §16.16 and Ch.16

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).

16.15 Integral Representations and Integrals 16.17 Definition

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