Formula:KLS:14.10:13

<< Formula:KLS:14.10:12

formula in Continuous q-Jacobi

Formula:KLS:14.10:14 >>

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\!\left(x|q\right)=\frac{2^{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\left(q^{n+\alpha+\beta+1};q\right)_{n}}{\left(q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)};q\right)_{n}}{\widehat{P}}^{(\alpha,\beta)}_{n}\!\left(x\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqJacobi
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{\widehat{P}}^{(\alpha,\beta)}_{n}}}}$ : monic continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsqJacobi

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:12

formula in Continuous q-Jacobi

Formula:KLS:14.10:14 >>