Formula:KLS:14.12:03

<< Formula:KLS:14.12:02

formula in Little q-Jacobi

Formula:KLS:14.12:06 >>

${\displaystyle{\displaystyle{\displaystyle-xp_{n}\!\left(x;a,b;q\right)=A_{n}p_{n+1}\!\left(x;a,b;q\right)-\left(A_{n}+C_{n}\right)p_{n}\!\left(x;a,b;q\right){}+C_{n}p_{n-1}\!\left(x;a,b;q\right)}}}$

Substitution(s)

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

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi

Bibliography

Equation in Section 14.12 of KLS.

URL links

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<< Formula:KLS:14.12:02

formula in Little q-Jacobi

Formula:KLS:14.12:06 >>