Formula:KLS:14.05:70

<< Formula:KLS:14.05:67

formula in Big q-Jacobi: Special case

Formula:KLS:14.05:71 >>

${\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x;c;q\right)={\widehat{P}}_{n+1}\!\left(x;c;q\right)+\left[1-(A_{n}+C_{n})\right]{\widehat{P}}_{n}\!\left(x;c;q\right)+A_{n-1}C_{n}{\widehat{P}}_{n-1}\!\left(x;c;q\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle C_{n}=-cq^{n+1}\frac{(1-q^{n})(1-c^{-1}q^{n})}{(1+q^{n})(1-q^{2n+1})}}}}$ &
${\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(1-q^{n+1})(1-cq^{n+1})}{(1+q^{n+1})(1-q^{2n+1})}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle{\widehat{P}}_{n}}}}$ : monic big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:monicbigqLegendre

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:67

formula in Big q-Jacobi: Special case

Formula:KLS:14.05:71 >>