Formula:KLS:14.05:19

<< Formula:KLS:14.05:18

formula in Big q-Jacobi

Formula:KLS:14.05:20 >>

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{cqx^{-1},0}{cq}{q}{xt}\,\qHyperrphis{1}{1}@@{bc^{-1}x}{abc^{-1}q}{q}{aqt}{}=\sum_{n=0}^{\infty}\frac{\left(aq;q\right)_{n}}{\left(abc^{-1}q,q;q\right)_{n}}P_{n}\!\left(x;a,b,c;q\right)t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:18

formula in Big q-Jacobi

Formula:KLS:14.05:20 >>