Formula:KLS:14.05:53

<< Formula:KLS:14.05:52

formula in Big q-Jacobi

Formula:KLS:14.05:54 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,a,1,1;q\right)=\frac{\left(q;q\right)_{n}}{\left(qa^{2};q\right)_{n}}(qa)^{n}\sum_{k=0}^{[\frac{1}{2}n]}(-1)^{k}q^{k(k-1)}\frac{\left(qa^{2};q^{2}\right)_{n-k}}{\left(q^{2};q^{2}\right)_{k}\left(q;q\right)_{n-2k}}x^{n-2k}}}}$

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:14.05:52

formula in Big q-Jacobi

Formula:KLS:14.05:54 >>