Formula:KLS:14.05:42

<< Formula:KLS:14.05:41

formula in Big q-Jacobi

Formula:KLS:14.05:43 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,c,d;q\right)=(qac^{-1}x)^{n}\frac{\left(qb,q;q\right)_{n}}{\left(-qac^{-1}d;q\right)_{n}}\sum_{k=0}^{n}\frac{\left(cx^{-1};q\right)_{n-k}}{\left(q,qa;q\right)_{n-k}}\frac{\left(-qbd^{-1}x;q\right)_{k}}{\left(qb,q;q\right)_{k}}(-1)^{k}q^{\frac{1}{2}k(k-1)}(-dx^{-1})^{k}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial with four parameters : http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:41

formula in Big q-Jacobi

Formula:KLS:14.05:43 >>