# Β§17.11 Transformations of $q$-Appell Functions

 17.11.1 $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx,b^{\prime}y;q% \right)_{\infty}}{\left(c,x,y;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({c/a,x,y% \atop bx,b^{\prime}y};q,a\right),$ β Symbols: $\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$: first $q$-Appell function, ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: 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: Β§17.11, Erratum (V1.0.10) for Section 17.1 Permalink: http://dlmf.nist.gov/17.11.E1 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for $\Phi^{(1)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.11 and Ch.17
 17.11.2 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}},$ β Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammerβs symbol (or shifted factorial), $\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$: 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: Erratum (V1.0.10) for Section 17.1, Erratum (V1.1.3) for Equation (17.11.2) Permalink: http://dlmf.nist.gov/17.11.E2 Encodings: TeX, pMML, png Correction (effective with 1.1.3): The factor ${\left(q\right)_{r}}$ originally used in the denominator has been corrected to be $\left(q;q\right)_{r}$. Correction (effective with 1.0.10): The notation for $\Phi^{(2)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.11 and Ch.17
 17.11.3 $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)=\frac{\left(a,bx;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a^{% \prime},b^{\prime};q\right)_{n}\left(x;q\right)_{r}\left(c/a;q\right)_{n+r}a^{% r}y^{n}}{\left(q,c/a;q\right)_{n}\left(q,bx;q\right)_{r}}.$ β Symbols: $\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$: 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: Β§17.11, Erratum (V1.0.10) for Section 17.1 Permalink: http://dlmf.nist.gov/17.11.E3 Encodings: TeX, pMML, png Correction (effective with 1.0.10): The notation for $\Phi^{(3)}$ has been updated to that of Gasper and Rahman (2004) to explicitly include the $q$ argument. See also: Annotations for Β§17.11 and Ch.17

Of (17.11.1)β(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 $\sum_{m_{1},\dots,m_{n}\geqq 0}\frac{\left(a;q\right)_{m_{1}+m_{2}+\cdots+m_{n% }}\left(b_{1};q\right)_{m_{1}}\left(b_{2};q\right)_{m_{2}}\cdots\left(b_{n};q% \right)_{m_{n}}x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}}{\left(q;q\right% )_{m_{1}}\left(q;q\right)_{m_{2}}\cdots\left(q;q\right)_{m_{n}}\left(c;q\right% )_{m_{1}+m_{2}+\cdots+m_{n}}}=\frac{\left(a,b_{1}x_{1},b_{2}x_{2},\dots,b_{n}x% _{n};q\right)_{\infty}}{\left(c,x_{1},x_{2},\dots,x_{n};q\right)_{\infty}}{{}_% {n+1}\phi_{n}}\left({c/a,x_{1},x_{2},\dots,x_{n}\atop b_{1}x_{1},b_{2}x_{2},% \dots,b_{n}x_{n}};q,a\right).$