Basic hypergeometric functions
Basic hypergeometric functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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. }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 }}