§17.11 Transformations of $q$ -Appell Functions

Notes:: See Andrews (1972).
Keywords:: $q$ -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 $q$ -Appell function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base, $x$ : real variable and $y$ : 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 $q$ -Appell function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $x$ : real variable and $y$ : 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 $q$ -Appell function, $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -shifted factorial), $q$ : complex base, $n$ : nonnegative integer, $r$ : nonnegative integer, $x$ : real variable and $y$ : 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 $n=2$ .

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:: $\left(a;q\right)_{{n}}$ : $q$ -factorial (or $q$ -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 $q$ -hypergeometric) function, $q$ : complex base, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/17.11.E4
Encodings:: TeX, pMML, png

§17.11 Transformations of -Appell Functions

§17.11 Transformations of $q$ -Appell Functions