Formula:KLS:14.08:35

<< Formula:KLS:14.08:34

formula in Al-Salam-Chihara

Formula:KLS:14.08:36 >>

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

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
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:34

formula in Al-Salam-Chihara

Formula:KLS:14.08:36 >>