-Hypergeometric and Related Functions Properties 17.10 Transformations of $\mathop{{{}_{{r}}\psi_{{r}}}\/}\nolimits$ Functions 17.12 Bailey Pairs

§17.11 Transformations of -Appell Functions

Notes:: See Andrews (1972).
Keywords:: -Appell functions
Permalink:: http://dlmf.nist.gov/17.11

17.11.1 $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)=\frac{% \left(a,bx,b^{{\prime}}y;q\right)_{{\infty}}}{\left(c,x,y;q\right)_{{\infty}}}% \mathop{{{}_{{3}}\phi_{{2}}}\/}\nolimits\!\left({c/a,x,y\atop bx,b^{{\prime}}y% };q,a\right),$

Symbols:: $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ : first -Appell function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r+1}}\phi_{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}% \atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base, : real variable and : real variable
Referenced by:: §17.11
Permalink:: http://dlmf.nist.gov/17.11.E1
Encodings:: TeX, pMML, png

17.11.2 $\mathop{\Phi^{{(2)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c,c^{{\prime}};x,y% \right)=\frac{\left(b,ax;q\right)_{{\infty}}}{\left(c,x;q\right)_{{\infty}}}% \sum_{{n,r\geqq 0}}\frac{\left(a,b^{{\prime}};q\right)_{{n}}\left(c/b,x;q% \right)_{{r}}b^{r}y^{n}}{\left(q,c^{{\prime}};q\right)_{{n}}\left(q\right)_{{r% }}\left(ax;q\right)_{{n+r}}},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ : first -Appell function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer, : real variable and : real variable
Permalink:: http://dlmf.nist.gov/17.11.E2
Encodings:: TeX, pMML, png

17.11.3 $\mathop{\Phi^{{(3)}}\/}\nolimits\!\left(a,a^{{\prime}};b,b^{{\prime}};c;x,y% \right)=\frac{\left(a,bx;q\right)_{{\infty}}}{\left(c,x;q\right)_{{\infty}}}% \sum_{{n,r\geqq 0}}\frac{\left(a^{{\prime}},b^{{\prime}};q\right)_{{n}}\left(x% ;q\right)_{{r}}\left(c/a;q\right)_{{n+r}}a^{r}y^{n}}{\left(q,c/a;q\right)_{{n}% }\left(q,bx;q\right)_{{r}}}.$

Symbols:: $\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$ : first -Appell function, $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), : complex base, : nonnegative integer, : nonnegative integer, : real variable and : real variable
Referenced by:: §17.11
Permalink:: http://dlmf.nist.gov/17.11.E3
Encodings:: TeX, pMML, png

Of (17.11.1)–(17.11.3) only (17.11.1) has a natural generalization: the following sum reduces to (17.11.1) when .

17.11.4 $\sum_{{m_{1},\dots,m_{n}\geqq 0}}\frac{\left(a;q\right)_{{m_{1}+m_{2}+\cdots+m% _{n}}}\left(b_{1};q\right)_{{m_{1}}}\left(b_{2};q\right)_{{m_{2}}}\cdots\left(% b_{n};q\right)_{{m_{n}}}x_{1}^{{m_{1}}}x_{2}^{{m_{2}}}\cdots x_{n}^{{m_{n}}}}{% \left(q;q\right)_{{m_{1}}}\left(q;q\right)_{{m_{2}}}\cdots\left(q;q\right)_{{m% _{n}}}\left(c;q\right)_{{m_{1}+m_{2}+\cdots+m_{n}}}}=\frac{\left(a,b_{1}x_{1},% b_{2}x_{2},\dots,b_{n}x_{n};q\right)_{{\infty}}}{\left(c,x_{1},x_{2},\dots,x_{% n};q\right)_{{\infty}}}\mathop{{{}_{{n+1}}\phi_{{n}}}\/}\nolimits\!\left({c/a,% x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x_{n}};q,a\right).$

Symbols:: $\left(a;q\right)_{{n}}$ : -factorial (or -shifted factorial), $\mathop{{{}_{{r+1}}\phi_{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}% \atop b_{1},b_{2},\dots,b_{s}};q,z\right)$ : basic hypergeometric (or -hypergeometric) function, : complex base, : nonnegative integer, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/17.11.E4
Encodings:: TeX, pMML, png

© 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. 17.10 Transformations of $\mathop{{{}_{{r}}\psi_{{r}}}\/}\nolimits$ Functions 17.12 Bailey Pairs