Formula:KLS:14.08:36

<< Formula:KLS:14.08:35

formula in Al-Salam-Chihara

Formula:KLS:14.08:37 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{(1-q^{2x}a^{-2})\left(a^{-2},(ab)^{-1};q\right)_{x}}{(1-a^{-2})\left(q,bqa^{-1};q\right)_{x}}(ba^{-1})^{x}q^{x^{2}}Q_{m}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)=\frac{\left(qa^{-2};q\right)_{\infty}}{\left(ba^{-1}q;q\right)_{\infty}}\left(q,(ab)^{-1};q\right)_{n}(ab)^{n}q^{-n^{2}}\delta_{m,n}(ab>1,\;qb<a)}}}$

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
${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-inverse Al-Salam-Chihara polynomial : http://dlmf.nist.gov/23.1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:35

formula in Al-Salam-Chihara

Formula:KLS:14.08:37 >>