Formula:KLS:14.22:20

<< Formula:KLS:14.22:19

formula in q-Bessel

Formula:KLS:14.22:21 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\left(t;q\right)_{\infty}}{\left(xt;q\right)_{\infty}}\,\qHyperrphis{1}{3}@@{xt}{0,0,t}{q}{-aqxt}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q;q\right)_{n}}y_{n}\!\left(x;a;q\right)t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle y_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Bessel polynomial : http://drmf.wmflabs.org/wiki/Definition:qBessel

Bibliography

Equation in Section 14.22 of KLS.

URL links

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<< Formula:KLS:14.22:19

formula in q-Bessel

Formula:KLS:14.22:21 >>