Some q-analogues of special functions

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Some q-analogues of special functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{q-Exponential function@$q$-Exponential function} \qexpKLS{q}@{z}:=\qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^n}{\qPochhammer{q}{q}{n}} =\frac{1}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1 }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle |z|<1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qExpKLS{q}@{z}:=\qHyperrphis{0}{0}@@{-}{-}{q}{-z}= \sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}z^n=\qPochhammer{-z}{q}{\infty} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 0<|q|<1}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qexpKLS{q}@{z)\E_q(-z}=1 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}\qexpKLS{q}@{(1-q)z}=\lim\limits_{q\rightarrow 1}\qExpKLS{q}@{(1-q)z}=\expe^z }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qcosKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)+{\mathrm e}_q(-\iunit z}}{2}= \sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{\qPochhammer{q}{q}{2n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qsinKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)-{\mathrm e}_q(-\iunit z}}{2\iunit}= \sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{\qPochhammer{q}{q}{2n+1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qCosKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}+\qExpKLS{q}@{-\iunit z}}{2} =\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n}{2}}z^{2n}}{\qPochhammer{q}{q}{2n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qSinKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}-\qExpKLS{q}@{-\iunit z}}{2\iunit} =\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n+1}{2}}z^{2n+1}}{\qPochhammer{q}{q}{2n+1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qexpKLS{q}@{\iunit z}=\qcosKLS{q}@{z}+\iunit\qsinKLS{q}@{z}\quad\textrm{and}\quad \qExpKLS{q}@{\iunit z}=\qCosKLS{q}@{z}+\iunit\:\qSinKLS{q}@{z} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qcosKLS{q}@{z}\qCosKLS{q}@{z}+\qsinKLS{q}@{z}\qSinKLS{q}@{z}=1 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qsinKLS{q}@{z}\qCosKLS{q}@{z}-\qcosKLS{q}@{z}\qSinKLS{q}@{z}=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \index{q-Bessel function@$q$-Bessel function}\index{Jackson's q-Bessel function@Jackson's $q$-Bessel function} \JacksonqBesselI{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{2}{1}@@{0,0}{q^{\nu+1}}{q}{-\frac{z^2}{4}} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle |z|<2}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \JacksonqBesselII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} \left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^2}{4}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \JacksonqBesselII{\nu}@{z}{q}=\qPochhammer{-\frac{z^2}{4}}{q}{\infty}\cdot \JacksonqBesselI{\nu}@{z}{q} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle |z|<2}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}J_{\nu}^{(k)}((1-q)z;q)=\BesselJ{\nu}@{z} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle k=1,2}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \JacksonqBesselIII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}} z^{\nu}\,\qHyperrphis{1}{1}@@{0}{q^{\nu+1}}{q}{qz^2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim\limits_{q\rightarrow 1}\JacksonqBesselIII{\nu}@{(1-q)z}{q}=\BesselJ{\nu}@{2z} }}

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