Hypergeometric functions
Hypergeometric functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{Hypergeometric function} \HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}:= \sum\limits_{k=0}^{\infty}\frac{\pochhammer{a_1,\ldots,a_r}{k}}{\pochhammer{b_1,\ldots,b_s}{k}} \frac{z^k}{k!} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pochhammer{a_1,\ldots,a_r}{k}:=\pochhammer{a_1}{k}\cdots\pochhammer{a_r}{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 \HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z} =\HyperpFq{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}} =\HyperpFq{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{\lambda\rightarrow\infty} \HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}= \HyperpFq{r}{s-1}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\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} \HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z} =\HyperpFq{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{N}@{x}{\alpha}{\beta}{N}=\sum_{k=0}^N\frac{\pochhammer{N+\alpha+\beta+1}{k}\pochhammer{-x}{k}}{\pochhammer{\alpha+1}{k}k!} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left[f(t)\right]_N:=\sum_{k=0}^N\frac{f^{(k)}(0)}{k!}t^k }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^t\,\HyperpFq{1}{1}@@{-x}{-N}{-\frac{t}{p}}= \sum_{k=0}^{\infty}\frac{t^k}{k!}\sum_{m=0}^x\frac{\pochhammer{-x}{m}}{\pochhammer{-N}{m}m!} \left(-\frac{t}{p}\right)^m }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^N\frac{\Krawtchouk{n}@{x}{p}{N}}{n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^z=\HyperpFq{0}{0}@@{-}{-}{z} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \cos@@{z}=\HyperpFq{0}{1}@@{-}{\frac{1}{2}}{-\frac{z^2}{4}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sin@@{z}=z\,\HyperpFq{0}{1}@@{-}{\frac{3}{2}}{-\frac{z^2}{4}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{Bessel function} \BesselJ{\nu}@{z}:=\frac{\left(\frac{1}{2}z\right)^{\nu}}{\EulerGamma@{\nu+1}}\ \HyperpFq{0}{1}@@{-}{\nu+1}{-\frac{z^2}{4}} }}