Formula:KLS:14.22:09

<< Formula:KLS:14.22:08

formula in q-Bessel

Formula:KLS:14.22:10 >>

${\displaystyle{\displaystyle{\displaystyle y_{n}\!\left(x;a;q\right)=(-1)^{n}q^{-\binomial{n}{2}}\left(-aq^{n};q\right)_{n}{\widehat{y}}_{n}\!\left(x\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle y_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Bessel polynomial : http://drmf.wmflabs.org/wiki/Definition:qBessel
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{\widehat{y}}_{n}}}}$ : monic ${\displaystyle{\displaystyle{\displaystyle q}}}$-Bessel polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqBessel

Bibliography

Equation in Section 14.22 of KLS.

URL links

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<< Formula:KLS:14.22:08

formula in q-Bessel

Formula:KLS:14.22:10 >>