Formula:KLS:14.19:15

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formula in Continuous q-Laguerre

Formula:KLS:14.19:16 >>

${\displaystyle{\displaystyle{\displaystyle\delta_{q}P^{(\alpha)}_{n}\!\left(x|q\right)=-q^{-n+\frac{1}{2}\alpha+\frac{3}{4}}({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})P^{(\alpha+1)}_{n-1}\!\left(x|q\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P^{(n)}_{\alpha}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLaguerre
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.19 of KLS.

URL links

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formula in Continuous q-Laguerre

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