16 Generalized Hypergeometric Functions and Meijer

-Function Two-Variable Hypergeometric Functions 16.15 Integral Representations and Integrals 16.17 Definition

§16.16 Transformations of Variables

Keywords:: Appell functions
Permalink:: http://dlmf.nist.gov/16.16

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

§16.16(i) Reduction Formulas

Keywords:: Appell functions
Permalink:: http://dlmf.nist.gov/16.16.i

16.16.1 $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\beta+\beta% ^{{\prime}};x,y\right)=(1-y)^{{-\alpha}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits% \!\left({\alpha,\beta\atop\beta+\beta^{{\prime}}};\frac{x-y}{1-y}\right),$

Symbols:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E1
Encodings:: TeX, pMML, png

16.16.2 $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,% \beta^{{\prime}};x,y\right)=(1-y)^{{-\alpha}}\mathop{{{}_{{2}}F_{{1}}}\/}% \nolimits\!\left({\alpha,\beta\atop\gamma};\frac{x}{1-y}\right),$

Symbols:: $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,% \gamma^{{\prime}};x,y\right)$ : Appell function and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E2
Encodings:: TeX, pMML, png

16.16.3 $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,% \alpha;x,y\right)=(1-y)^{{-\beta^{{\prime}}}}\mathop{{F_{{1}}}\/}\nolimits\!% \left(\beta;\alpha-\beta^{{\prime}},\beta^{{\prime}};\gamma;x,\frac{x}{1-y}% \right),$

Symbols:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function and $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,% \gamma^{{\prime}};x,y\right)$ : Appell function
Permalink:: http://dlmf.nist.gov/16.16.E3
Encodings:: TeX, pMML, png

16.16.4 $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\gamma-\alpha;\beta,\beta^{{\prime% }};\gamma;x,y\right)=(1-y)^{{-\beta^{{\prime}}}}\mathop{{F_{{1}}}\/}\nolimits% \!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,\frac{y}{y-1}\right),$

Symbols:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function and $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{% \prime}};\gamma;x,y\right)$ : Appell function
Permalink:: http://dlmf.nist.gov/16.16.E4
Encodings:: TeX, pMML, png

16.16.5 $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\gamma-\alpha;\beta,\gamma-\beta;% \gamma;x,y\right)=(1-y)^{{\alpha+\beta-\gamma}}\mathop{{{}_{{2}}F_{{1}}}\/}% \nolimits\!\left({\alpha,\beta\atop\gamma};x+y-xy\right),$

Symbols:: $\mathop{{F_{{3}}}\/}\nolimits\!\left(\alpha,\alpha^{{\prime}};\beta,\beta^{{% \prime}};\gamma;x,y\right)$ : Appell function and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function
Permalink:: http://dlmf.nist.gov/16.16.E5
Encodings:: TeX, pMML, png

16.16.6 $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\alpha+\beta-\gamma+1% ;x(1-y),y(1-x)\right)=\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({\alpha,% \beta\atop\gamma};x\right)\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({\alpha% ,\beta\atop\alpha+\beta-\gamma+1};y\right).$

Symbols:: $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function and $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function
Referenced by:: §16.16(i)
Permalink:: http://dlmf.nist.gov/16.16.E6
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol and : : factorial
Permalink:: http://dlmf.nist.gov/16.16.E7
Encodings:: TeX, pMML, png

see Burchnall and Chaundy (1940, 1941).

§16.16(ii) Other Transformations

Notes:: See Erdélyi et al. (1953a, §5.11).
Keywords:: Appell functions, generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.16.ii

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:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function
Permalink:: http://dlmf.nist.gov/16.16.E8
Encodings:: TeX, pMML, png

16.16.9 $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,% \gamma^{{\prime}};x,y\right)=(1-x)^{{-\alpha}}\mathop{{F_{{2}}}\/}\nolimits\!% \left(\alpha;\gamma-\beta,\beta^{{\prime}};\gamma,\gamma^{{\prime}};\frac{x}{x% -1},\frac{y}{1-x}\right),$

Symbols:: $\mathop{{F_{{2}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma,% \gamma^{{\prime}};x,y\right)$ : Appell function
Permalink:: http://dlmf.nist.gov/16.16.E9
Encodings:: TeX, pMML, png

16.16.10 $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y% \right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma^{{\prime}}\right)% \mathop{\Gamma\/}\nolimits\!\left(\beta-\alpha\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\gamma^{{\prime}}-\alpha\right)\mathop{\Gamma\/}\nolimits\!% \left(\beta\right)}(-y)^{{-\alpha}}\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha% ;\alpha-\gamma^{{\prime}}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\mathop{\Gamma\/}\nolimits\!\left(\gamma^{{\prime}}\right)% \mathop{\Gamma\/}\nolimits\!\left(\alpha-\beta\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\gamma^{{\prime}}-\beta\right)\mathop{\Gamma\/}\nolimits\!% \left(\alpha\right)}(-y)^{{-\beta}}\mathop{{F_{{4}}}\/}\nolimits\!\left(\beta;% \beta-\gamma^{{\prime}}+1;\gamma,\beta-\alpha+1;\frac{x}{y},\frac{1}{y}\right).$

Symbols:: $\mathop{{F_{{4}}}\/}\nolimits\!\left(\alpha;\beta;\gamma,\gamma^{{\prime}};x,y\right)$ : Appell function and $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
Permalink:: http://dlmf.nist.gov/16.16.E10
Encodings:: TeX, pMML, png

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 16.15 Integral Representations and Integrals 16.17 Definition

§16.16 Transformations of Variables

Contents

§16.16(i) Reduction Formulas

§16.16(ii) Other Transformations