Formula:KLS:14.08:34

<< Formula:KLS:14.08:33

formula in Al-Salam-Chihara

Formula:KLS:14.08:35 >>

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)=(-ab^{-1})^{x}q^{-\frac{1}{2}x(x+1)}\frac{\left(qba^{-1};q\right)_{x}}{\left(a^{-1}b^{-1};q\right)_{x}}\qHyperrphis{2}{1}@@{q^{-x},a^{-2}q^{x}}{qba^{-1}}{q}{q^{n+1}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-inverse Al-Salam-Chihara polynomial : http://dlmf.nist.gov/23.1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:33

formula in Al-Salam-Chihara

Formula:KLS:14.08:35 >>