Formula:KLS:14.08:28

<< Formula:KLS:14.08:27

formula in Al-Salam-Chihara

Formula:KLS:14.08:29 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma{\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab,\gamma{\mathrm{e}^{\mathrm{i}\theta}}t}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(\gamma;q\right)_{n}}{\left(ab,q;q\right)_{n}}Q_{n}\!\left(x;a,b\,|\,q\right)t^{n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$ &
${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.8 of KLS.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:KLS:14.08:27

formula in Al-Salam-Chihara

Formula:KLS:14.08:29 >>