Ï s r â¡ ( a 1 , ⦠, a r b 1 , ⦠, b s ; q , z ) := â k = 0 â \qPochhammer ⢠a 1 , ⦠, a r ⢠q ⢠k \qPochhammer ⢠b 1 , ⦠, b s ⢠q ⢠k ⢠( - 1 ) ( 1 + s - r ) ⢠k ⢠q ( 1 + s - r ) ⢠\binomial ⢠k ⢠2 ⢠z k \qPochhammer ⢠q ⢠q ⢠k assign q-hypergeometric-rphis ð ð subscript ð 1 ⦠subscript ð ð subscript ð 1 ⦠subscript ð ð ð ð§ superscript subscript ð 0 \qPochhammer subscript ð 1 ⦠subscript ð ð ð ð \qPochhammer subscript ð 1 ⦠subscript ð ð ð ð superscript 1 1 ð ð ð superscript ð 1 ð ð \binomial ð 2 superscript ð§ ð \qPochhammer ð ð ð {\displaystyle{\displaystyle{\displaystyle{}{{}_{r}\phi_{s}}\!\left({a_{1},% \ldots,a_{r}\atop b_{1},\ldots,b_{s}};q,z\right){}:=\sum\limits_{k=0}^{\infty}% \frac{\qPochhammer{a_{1},\ldots,a_{r}}{q}{k}}{\qPochhammer{b_{1},\ldots,b_{s}}% {q}{k}}(-1)^{(1+s-r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^{k}}{\qPochhammer{q}{q% }{k}}}}} {\displaystyle \index{Basic hypergeometric function} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{z} {}:=\sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_r}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}} (-1)^{(1+s-r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^k}{\qPochhammer{q}{q}{k}} }
Ï = { â if â r < s + 1 1 if â r = s + 1 0 if â r > s + 1 . ð cases if ð ð 1 1 if ð ð 1 0 if ð ð 1 {\displaystyle{\displaystyle{\displaystyle\rho=\left\{\begin{array}[]{ll}% \displaystyle\infty&\quad\textrm{if}\quad r<s+1\\ \displaystyle 1&\quad\textrm{if}\quad r=s+1\\ \displaystyle 0&\quad\textrm{if}\quad r>s+1.\end{array}\right.}}} {\displaystyle \rho=\left\{\begin{array}{ll} \displaystyle \infty & \quad\textrm{if}\quad r < s+1\[5mm] \displaystyle 1 & \quad\textrm{if}\quad r = s+1\[5mm] \displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right. } Ï s s + 1 â¡ ( a 1 , ⦠, a s + 1 b 1 , ⦠, b s ; q , z ) = â k = 0 â \qPochhammer ⢠a 1 , ⦠, a s + 1 ⢠q ⢠k \qPochhammer ⢠b 1 , ⦠, b s ⢠q ⢠k ⢠z k \qPochhammer ⢠q ⢠q ⢠k q-hypergeometric-rphis ð 1 ð subscript ð 1 ⦠subscript ð ð 1 subscript ð 1 ⦠subscript ð ð ð ð§ superscript subscript ð 0 \qPochhammer subscript ð 1 ⦠subscript ð ð 1 ð ð \qPochhammer subscript ð 1 ⦠subscript ð ð ð ð superscript ð§ ð \qPochhammer ð ð ð {\displaystyle{\displaystyle{\displaystyle{{}_{s+1}\phi_{s}}\!\left({a_{1},% \ldots,a_{s+1}\atop b_{1},\ldots,b_{s}};q,z\right)=\sum\limits_{k=0}^{\infty}% \frac{\qPochhammer{a_{1},\ldots,a_{s+1}}{q}{k}}{\qPochhammer{b_{1},\ldots,b_{s% }}{q}{k}}\frac{z^{k}}{\qPochhammer{q}{q}{k}}}}} {\displaystyle \qHyperrphis{s+1}{s}@@{a_1,\ldots,a_{s+1}}{b_1,\ldots,b_s}{q}{z}= \sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_{s+1}}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}} \frac{z^k}{\qPochhammer{q}{q}{k}} }
lim q â 1 â¡ Ï s r â¡ ( q a 1 , ⦠, q a r q b 1 , ⦠, q b s ; q , ( q - 1 ) 1 + s - r ⢠z ) = F s r â¡ ( a 1 , ⦠, a r b 1 , ⦠, b s ; z ) subscript â ð 1 q-hypergeometric-rphis ð ð superscript ð subscript ð 1 ⦠superscript ð subscript ð ð superscript ð subscript ð 1 ⦠superscript ð subscript ð ð ð superscript ð 1 1 ð ð ð§ Gauss-hypergeometric-pFq ð ð subscript ð 1 ⦠subscript ð ð subscript ð 1 ⦠subscript ð ð ð§ {\displaystyle{\displaystyle{\displaystyle\lim\limits_{q\rightarrow 1}{{}_{r}% \phi_{s}}\!\left({q^{a_{1}},\ldots,q^{a_{r}}\atop q^{b_{1}},\ldots,q^{b_{s}}};% q,(q-1)^{1+s-r}z\right)={{}_{r}F_{s}}\!\left({a_{1},\ldots,a_{r}\atop b_{1},% \ldots,b_{s}};z\right)}}} {\displaystyle \lim\limits_{q\rightarrow 1} \qHyperrphis{r}{s}@@{q^{a_1},\ldots,q^{a_r}}{q^{b_1},\ldots,q^{b_s}}{q}{(q-1)^{1+s-r}z} =\HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z} } lim a r â â â¡ Ï s r â¡ ( a 1 , ⦠, a r b 1 , ⦠, b s ; q , z a r ) = Ï s r - 1 â¡ ( a 1 , ⦠, a r - 1 b 1 , ⦠, b s ; q , z ) subscript â subscript ð ð q-hypergeometric-rphis ð ð subscript ð 1 ⦠subscript ð ð subscript ð 1 ⦠subscript ð ð ð ð§ subscript ð ð q-hypergeometric-rphis ð 1 ð subscript ð 1 ⦠subscript ð ð 1 subscript ð 1 ⦠subscript ð ð ð ð§ {\displaystyle{\displaystyle{\displaystyle\lim\limits_{a_{r}\rightarrow\infty}% {{}_{r}\phi_{s}}\!\left({a_{1},\ldots,a_{r}\atop b_{1},\ldots,b_{s}};q,\frac{z% }{a_{r}}\right)={{}_{r-1}\phi_{s}}\!\left({a_{1},\ldots,a_{r-1}\atop b_{1},% \ldots,b_{s}};q,z\right)}}} {\displaystyle \lim\limits_{a_r\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{\frac{z}{a_r}}= \qHyperrphis{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{q}{z} } Ï s r â¡ ( a 1 , ⦠, a r - 1 , μ b 1 , ⦠, b s - 1 , μ ; q , z ) = Ï s - 1 r - 1 â¡ ( a 1 , ⦠, a r - 1 b 1 , ⦠, b s - 1 ; q , z ) q-hypergeometric-rphis ð ð subscript ð 1 ⦠subscript ð ð 1 ð subscript ð 1 ⦠subscript ð ð 1 ð ð ð§ q-hypergeometric-rphis ð 1 ð 1 subscript ð 1 ⦠subscript ð ð 1 subscript ð 1 ⦠subscript ð ð 1 ð ð§ {\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}\!\left({a_{1},% \ldots,a_{r-1},\mu\atop b_{1},\ldots,b_{s-1},\mu};q,z\right)={{}_{r-1}\phi_{s-% 1}}\!\left({a_{1},\ldots,a_{r-1}\atop b_{1},\ldots,b_{s-1}};q,z\right)}}} {\displaystyle \qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{q}{z}= \qHyperrphis{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{q}{z} } lim λ â â â¡ Ï s r â¡ ( a 1 , ⦠, a r - 1 , λ ⢠a r b 1 , ⦠, b s ; q , z λ ) = Ï s r - 1 â¡ ( a 1 , ⦠, a r - 1 b 1 , ⦠, b s ; q , a r ⢠z ) subscript â ð q-hypergeometric-rphis ð ð subscript ð 1 ⦠subscript ð ð 1 ð subscript ð ð subscript ð 1 ⦠subscript ð ð ð ð§ ð q-hypergeometric-rphis ð 1 ð subscript ð 1 ⦠subscript ð ð 1 subscript ð 1 ⦠subscript ð ð ð subscript ð ð ð§ {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}{{}_{r}\phi_{s}}\!\left({a_{1},\ldots,a_{r-1},\lambda a_{r}\atop b_{1},% \ldots,b_{s}};q,\frac{z}{\lambda}\right)={{}_{r-1}\phi_{s}}\!\left({a_{1},% \ldots,a_{r-1}\atop b_{1},\ldots,b_{s}};q,a_{r}z\right)}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{q}{\frac{z}{\lambda}} =\qHyperrphis{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{q}{a_rz} } lim λ â â â¡ Ï s r â¡ ( a 1 , ⦠, a r b 1 , ⦠, b s - 1 , λ ⢠b s ; q , λ ⢠z ) = Ï s - 1 r â¡ ( a 1 , ⦠, a r b 1 , ⦠, b s - 1 ; q , z b s ) subscript â ð q-hypergeometric-rphis ð ð subscript ð 1 ⦠subscript ð ð subscript ð 1 ⦠subscript ð ð 1 ð subscript ð ð ð ð ð§ q-hypergeometric-rphis ð ð 1 subscript ð 1 ⦠subscript ð ð subscript ð 1 ⦠subscript ð ð 1 ð ð§ subscript ð ð {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}{{}_{r}\phi_{s}}\!\left({a_{1},\ldots,a_{r}\atop b_{1},\ldots,b_{s-1},% \lambda b_{s}};q,\lambda z\right)={{}_{r}\phi_{s-1}}\!\left({a_{1},\ldots,a_{r% }\atop b_{1},\ldots,b_{s-1}};q,\frac{z}{b_{s}}\right)}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{q}{\lambda z}= \qHyperrphis{r}{s-1}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{q}{\frac{z}{b_s}} } lim λ â â â¡ Ï s r â¡ ( a 1 , ⦠, a r - 1 , λ ⢠a r b 1 , ⦠, b s - 1 , λ ⢠b s ; q , z ) = Ï s - 1 r - 1 â¡ ( a 1 , ⦠, a r - 1 b 1 , ⦠, b s - 1 ; q , a r ⢠z b s ) subscript â ð q-hypergeometric-rphis ð ð subscript ð 1 ⦠subscript ð ð 1 ð subscript ð ð subscript ð 1 ⦠subscript ð ð 1 ð subscript ð ð ð ð§ q-hypergeometric-rphis ð 1 ð 1 subscript ð 1 ⦠subscript ð ð 1 subscript ð 1 ⦠subscript ð ð 1 ð subscript ð ð ð§ subscript ð ð {\displaystyle{\displaystyle{\displaystyle\lim\limits_{\lambda\rightarrow% \infty}{{}_{r}\phi_{s}}\!\left({a_{1},\ldots,a_{r-1},\lambda a_{r}\atop b_{1},% \ldots,b_{s-1},\lambda b_{s}};q,z\right)={{}_{r-1}\phi_{s-1}}\!\left({a_{1},% \ldots,a_{r-1}\atop b_{1},\ldots,b_{s-1}};q,\frac{a_{r}z}{b_{s}}\right)}}} {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{q}{z} =\qHyperrphis{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{q}{\frac{a_rz}{b_s}} } \qHahn ⢠N ⢠@ ⢠q - x ⢠α ⢠β ⢠N ⢠q = â k = 0 N \qPochhammer ⢠α ⢠β ⢠q N + 1 ⢠q ⢠k ⢠\qPochhammer ⢠q - x ⢠q ⢠k \qPochhammer ⢠α ⢠q ⢠q ⢠k ⢠\qPochhammer ⢠q ⢠q ⢠k ⢠q k \qHahn ð @ superscript ð ð¥ ð¼ ð½ ð ð superscript subscript ð 0 ð \qPochhammer ð¼ ð½ superscript ð ð 1 ð ð \qPochhammer superscript ð ð¥ ð ð \qPochhammer ð¼ ð ð ð \qPochhammer ð ð ð superscript ð ð {\displaystyle{\displaystyle{\displaystyle\qHahn{N}@{q^{-x}}{\alpha}{\beta}{N}% {q}=\sum_{k=0}^{N}\frac{\qPochhammer{\alpha\beta q^{N+1}}{q}{k}\qPochhammer{q^% {-x}}{q}{k}}{\qPochhammer{\alpha q}{q}{k}\qPochhammer{q}{q}{k}}q^{k}}}} {\displaystyle \qHahn{N}@{q^{-x}}{\alpha}{\beta}{N}{q}=\sum_{k=0}^N \frac{\qPochhammer{\alpha\beta q^{N+1}}{q}{k}\qPochhammer{q^{-x}}{q}{k}}{\qPochhammer{\alpha q}{q}{k}\qPochhammer{q}{q}{k}}q^k }