# §17.11 Transformations of $q$-Appell Functions

 17.11.1 $\mathop{\Phi^{(1)}\/}\nolimits\!\left(a;b,b^{\prime};c;q;x,y\right)=\frac{% \left(a,bx,b^{\prime}y;q\right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}% \mathop{{{}_{3}\phi_{2}}\/}\nolimits\!\left({c/a,x,y\atop bx,b^{\prime}y};q,a% \right),$ Symbols: $\mathop{\Phi^{(1)}\/}\nolimits\!\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};% \NVar{c};\NVar{q};\NVar{x},\NVar{y}\right)$: first $q$-Appell function, $\mathop{{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\/}\nolimits\!\left(\NVar{a_{0},\dots,% a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)$ or $\mathop{{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\/}\nolimits\!\left({\NVar{a_{0},\dots% ,a_{r}}\atop\NVar{b_{1},\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$-hypergeometric) function, $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: complex base, $x$: real variable and $y$: real variable Referenced by: §17.11, Other Changes Permalink: http://dlmf.nist.gov/17.11.E1 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for $\mathop{\Phi^{(1)}\/}\nolimits$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. Reported 2015-04-10 See also: Annotations for 17.11
 17.11.2 $\mathop{\Phi^{(2)}\/}\nolimits\!\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)% =\frac{\left(b,ax;q\right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r% \geqq 0}\frac{\left(a,b^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n% }}{\left(q,c^{\prime};q\right)_{n}{\left(q\right)_{r}}\left(ax;q\right)_{n+r}},$ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathop{\Phi^{(2)}\/}\nolimits\!\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};% \NVar{c},\NVar{c^{\prime}};\NVar{q};\NVar{x},\NVar{y}\right)$: second $q$-Appell function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable Referenced by: Other Changes Permalink: http://dlmf.nist.gov/17.11.E2 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for $\mathop{\Phi^{(2)}\/}\nolimits$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. Reported 2015-04-10 See also: Annotations for 17.11
 17.11.3 $\mathop{\Phi^{(3)}\/}\nolimits\!\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)% =\frac{\left(a,bx;q\right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r% \geqq 0}\frac{\left(a^{\prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}% \left(c/a;q\right)_{n+r}a^{r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right% )_{r}}.$ Symbols: $\mathop{\Phi^{(3)}\/}\nolimits\!\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},% \NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x},\NVar{y}\right)$: third $q$-Appell function, $\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}$: $q$-Pochhammer symbol (or $q$-shifted factorial), $\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}\right)_{\NVar{n}}$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $x$: real variable and $y$: real variable Referenced by: §17.11, Other Changes Permalink: http://dlmf.nist.gov/17.11.E3 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for $\mathop{\Phi^{(3)}\/}\nolimits$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. Reported 2015-04-10 See also: Annotations for 17.11

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