Formula:KLS:14.22:17

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formula in q-Bessel

Formula:KLS:14.22:18 >>

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)y_{n}\!\left(q^{x};a;q\right)=a^{n}(1-q)^{n}q^{n(n-1)}\left(\nabla_{q}\right)^{n}\left[w(x;aq^{2n};q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x}{2}}}{\left(q;q\right)_{x}}}}}$

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

Bibliography

Equation in Section 14.22 of KLS.

URL links

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formula in q-Bessel

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