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<p><b>New page</b></p><div>{{DISPLAYTITLE:Some ''q''-analogues of special functions}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Transformation formulas|Transformation formulas]] </div><br />
<div id="aligncenter"> [[Orthogonal_Polynomials#Some q-analogues of special functions|Some q-analogues of special functions]] </div><br />
<div id="alignright"> [[The q-derivative and q-integral|The q-derivative and q-integral]] >> </div><br />
</div><br />
<br />
== Some ''q''-analogues of special functions ==<br />
<br />
<math id="KLS:01.14:01">{\displaystyle <br />
\index{q-Exponential function@$q$-Exponential function}<br />
\qexpKLS{q}@{z}:=\qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^n}{\qPochhammer{q}{q}{n}}<br />
=\frac{1}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle |z|<1}</math></div><br /><br />
<math id="KLS:01.14:02">{\displaystyle <br />
\qExpKLS{q}@{z}:=\qHyperrphis{0}{0}@@{-}{-}{q}{-z}=<br />
\sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}z^n=\qPochhammer{-z}{q}{\infty}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle 0<|q|<1}</math></div><br /><br />
<math id="KLS:01.14:03">{\displaystyle <br />
\qexpKLS{q}@{z)\E_q(-z}=1<br />
}</math><br /><br />
<math id="KLS:01.14:04">{\displaystyle <br />
\lim\limits_{q\rightarrow 1}\qexpKLS{q}@{(1-q)z}=\lim\limits_{q\rightarrow 1}\qExpKLS{q}@{(1-q)z}=\expe^z<br />
}</math><br /><br />
<math id="KLS:01.14:05">{\displaystyle <br />
\qcosKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)+{\mathrm e}_q(-\iunit z}}{2}=<br />
\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{\qPochhammer{q}{q}{2n}}<br />
}</math><br /><br />
<math id="KLS:01.14:06">{\displaystyle <br />
\qsinKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)-{\mathrm e}_q(-\iunit z}}{2\iunit}=<br />
\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{\qPochhammer{q}{q}{2n+1}}<br />
}</math><br /><br />
<math id="KLS:01.14:07">{\displaystyle <br />
\qCosKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}+\qExpKLS{q}@{-\iunit z}}{2}<br />
=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n}{2}}z^{2n}}{\qPochhammer{q}{q}{2n}}<br />
}</math><br /><br />
<math id="KLS:01.14:08">{\displaystyle <br />
\qSinKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}-\qExpKLS{q}@{-\iunit z}}{2\iunit}<br />
=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n+1}{2}}z^{2n+1}}{\qPochhammer{q}{q}{2n+1}}<br />
}</math><br /><br />
<math id="KLS:01.14:09">{\displaystyle <br />
\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}<br />
}</math><br /><br />
<math id="KLS:01.14:10">{\displaystyle <br />
\qcosKLS{q}@{z}\qCosKLS{q}@{z}+\qsinKLS{q}@{z}\qSinKLS{q}@{z}=1<br />
}</math><br /><br />
<math id="KLS:01.14:11">{\displaystyle <br />
\qsinKLS{q}@{z}\qCosKLS{q}@{z}-\qcosKLS{q}@{z}\qSinKLS{q}@{z}=0<br />
}</math><br /><br />
<math id="KLS:01.14:12">{\displaystyle <br />
\index{q-Bessel function@$q$-Bessel function}\index{Jackson's q-Bessel function@Jackson's $q$-Bessel function}<br />
\JacksonqBesselI{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}}<br />
\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{2}{1}@@{0,0}{q^{\nu+1}}{q}{-\frac{z^2}{4}}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle |z|<2}</math></div><br /><br />
<math id="KLS:01.14:13">{\displaystyle <br />
\JacksonqBesselII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}}<br />
\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^2}{4}}<br />
}</math><br /><br />
<math id="KLS:01.14:14">{\displaystyle <br />
\JacksonqBesselII{\nu}@{z}{q}=\qPochhammer{-\frac{z^2}{4}}{q}{\infty}\cdot \JacksonqBesselI{\nu}@{z}{q}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle |z|<2}</math></div><br /><br />
<math id="KLS:01.14:15">{\displaystyle <br />
\lim\limits_{q\rightarrow 1}J_{\nu}^{(k)}((1-q)z;q)=\BesselJ{\nu}@{z}<br />
}</math><br />
<div align="right">Substitution(s): <math>{\displaystyle k=1,2}</math></div><br /><br />
<math id="KLS:01.14:16">{\displaystyle <br />
\JacksonqBesselIII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}}<br />
z^{\nu}\,\qHyperrphis{1}{1}@@{0}{q^{\nu+1}}{q}{qz^2}<br />
}</math><br /><br />
<math id="KLS:01.14:17">{\displaystyle <br />
\lim\limits_{q\rightarrow 1}\JacksonqBesselIII{\nu}@{(1-q)z}{q}=\BesselJ{\nu}@{2z}<br />
}</math><br />
<div id="drmf_foot"><br />
<div id="alignleft"> << [[Transformation formulas|Transformation formulas]] </div><br />
<div id="aligncenter"> [[Orthogonal_Polynomials#Some q-analogues of special functions|Some q-analogues of special functions]] </div><br />
<div id="alignright"> [[The q-derivative and q-integral|The q-derivative and q-integral]] >> </div><br />
</div></div>
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