Formula:KLS:14.12:02

<< Formula:KLS:14.12:01

formula in Little q-Jacobi

Formula:KLS:14.12:03 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\left(bq;q\right)_{k}}{\left(q;q\right)_{k}}(aq)^{k}p_{m}\!\left(q^{k};a,b;q\right)p_{n}\!\left(q^{k};a,b;q\right){}=\frac{\left(abq^{2};q\right)_{\infty}}{\left(aq;q\right)_{\infty}}\frac{(1-abq)(aq)^{n}}{(1-abq^{2n+1})}\frac{\left(q,bq;q\right)_{n}}{\left(aq,abq;q\right)_{n}}\,\delta_{m,n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.12 of KLS.

URL links

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<< Formula:KLS:14.12:01

formula in Little q-Jacobi

Formula:KLS:14.12:03 >>