Formula:KLS:14.10:65

<< Formula:KLS:14.10:64

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:66 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow 0}C_{n}\!\left(x;\beta\,|\,q\right)=\frac{H_{n}\!\left(x\,|\,q\right)}{\left(q;q\right)_{n}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle C_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqHermite
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:64

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:66 >>