Formula:KLS:14.08:07

<< Formula:KLS:14.08:04

formula in Al-Salam-Chihara

Formula:KLS:14.08:10 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}Q_{m}\!\left(x;a,b\,|\,q\right)Q_{n}\!\left(x;a,b\,|\,q\right)\,dx{}+\sum_{\begin{array}[]{c}\scriptstyle k\\ \scriptstyle 1<aq^{k}\leq a\end{array}}w_{k}Q_{m}\!\left(x_{k};a,b\,|\,q\right)Q_{n}\!\left(x_{k};a,b\,|\,q\right)=\frac{\,\delta_{m,n}}{\left(q^{n+1},abq^{n};q\right)_{\infty}}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{\infty}}{\left(q,ab,a^{-1}b;q\right)_{\infty}}\frac{(1-a^{2}q^{2k})\left(a^{2},ab;q\right)_{k}}{(1-a^{2})\left(q,ab^{-1}q;q\right)_{k}}q^{-k^{2}}\left(\frac{1}{a^{3}b}\right)^{k}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x_{k}=\frac{aq^{k}+\left(aq^{k}\right)^{-1}}{2}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle Q_{n}}}}$ : Al-Salam-Chihara polynomial : http://drmf.wmflabs.org/wiki/Definition:AlSalamChihara
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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
${\displaystyle{\displaystyle{\displaystyle\Pi}}}$ : product : http://drmf.wmflabs.org/wiki/Definition:prod
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.8 of KLS.

URL links

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<< Formula:KLS:14.08:04

formula in Al-Salam-Chihara

Formula:KLS:14.08:10 >>