Formula:KLS:14.12:34

<< Formula:KLS:14.12:31

formula in Little q-Jacobi: Special case

Formula:KLS:14.12:35 >>

${\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)={\widehat{p}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}){\widehat{p}}_{n}\!\left(x\right)+A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}}$

Substitution(s)

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

Proof

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Symbols List

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

Bibliography

Equation in Section 14.12 of KLS.

URL links

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<< Formula:KLS:14.12:31

formula in Little q-Jacobi: Special case

Formula:KLS:14.12:35 >>