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

§16.16(i) Reduction Formulas
§16.16(ii) Other Transformations

§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 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\beta+\beta^{\prime};x,y\right)=(1-y)% ^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\beta+\beta^{\prime}};\frac{x-y% }{1-y}\right),$

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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 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\beta^{\prime};x,y\right)=(1-y% )^{-\alpha}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};\frac{x}{1-y}\right),$

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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 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\alpha;x,y\right)=(1-y)^{-% \beta^{\prime}}{F_{1}}\left(\beta;\alpha-\beta^{\prime},\beta^{\prime};\gamma;% x,\frac{x}{1-y}\right),$

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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 ${F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,\frac{y}{y% -1}\right),$

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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 ${F_{3}}\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;\gamma;x,y\right)=(1-y)^{% \alpha+\beta-\gamma}{{}_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x+y-xy\right),$

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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 ${F_{4}}\left(\alpha,\beta;\gamma,\beta;x(1-y),y(1-x)\right)=(1-x)^{-\alpha}(1-% y)^{-\alpha}{F_{1}}\left(\alpha;\gamma-\beta,\alpha-\gamma+1;\gamma;\frac{x}{x% -1},\frac{xy}{(1-x)(1-y)}\right),$

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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 ${F_{4}}\left(\alpha,\beta;\gamma,\alpha+\beta-\gamma+1;x(1-y),y(1-x)\right)={{% }_{2}F_{1}}\left({\alpha,\beta\atop\gamma};x\right){{}_{2}F_{1}}\left({\alpha,% \beta\atop\alpha+\beta-\gamma+1};y\right).$

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

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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 )
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 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-x)^{-\beta}(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\gamma-\alpha;\beta,\beta^{\prime};\gamma;\frac% {x}{x-1},\frac{y}{y-1}\right)=(1-x)^{-\alpha}{F_{1}}\left(\alpha;\gamma-\beta-% \beta^{\prime},\beta^{\prime};\gamma;\frac{x}{x-1},\frac{y-x}{1-x}\right),$

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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 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=(1-% x)^{-\alpha}{F_{2}}\left(\alpha;\gamma-\beta,\beta^{\prime};\gamma,\gamma^{% \prime};\frac{x}{x-1},\frac{y}{1-x}\right),$

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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 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\frac{\Gamma\left(% \gamma^{\prime}\right)\Gamma\left(\beta-\alpha\right)}{\Gamma\left(\gamma^{% \prime}-\alpha\right)\Gamma\left(\beta\right)}(-y)^{-\alpha}{F_{4}}\left(% \alpha,\alpha-\gamma^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\Gamma\left(\gamma^{\prime}\right)\Gamma\left(\alpha-\beta\right% )}{\Gamma\left(\gamma^{\prime}-\beta\right)\Gamma\left(\alpha\right)}(-y)^{-% \beta}{F_{4}}\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-\alpha+1;\frac{x% }{y},\frac{1}{y}\right).$

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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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§16.16 Transformations of Variables

Contents

§16.16(i) Reduction Formulas

§16.16(ii) Other Transformations