Formula:KLS:14.10:08

<< Formula:KLS:14.10:02

formula in Continuous q-Jacobi

Formula:KLS:14.10:09 >>

${\displaystyle{\displaystyle{\displaystyle 2x{\tilde{P}}^{(n,\beta)}_{n}\!\left(x|q\right)=A_{n}{\tilde{P}}^{(n+1,\beta)}_{n}\!\left(x|q\right)+\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+q^{-\frac{1}{2}\alpha-\frac{1}{4}}-\left(A_{n}+C_{n}\right)\right]{\tilde{P}}^{(n,\beta)}_{n}\!\left(x|q\right){}+C_{n}{\tilde{P}}^{(n-1,\beta)}_{n}\!\left(x|q\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{n})(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})}{(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})}{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle{\tilde{P}}^{(\alpha,\beta)}_{n}}}}$ : normalized continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial ${\displaystyle{\displaystyle{\displaystyle{\tilde{P}}}}}$ : http://drmf.wmflabs.org/wiki/Definition:normctsqJacobiPtilde

Bibliography

Equation in Section 14.10 of KLS.

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<< Formula:KLS:14.10:02

formula in Continuous q-Jacobi

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