Formula:KLS:14.10:89

<< Formula:KLS:14.10:88

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:95 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x;1|q)}{\sqrt{1-x^{2}}}P_{m}\!\left(x|q\right)P_{n}\!\left(x|q\right)\,dx{}=\frac{\left(q^{\frac{1}{2}};q\right)_{\infty}}{\left(q,q,-q^{\frac{1}{2}},-q;q\right)_{\infty}}\frac{q^{\frac{1}{2}n}}{1-q^{n+\frac{1}{2}}}\,\delta_{m,n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;a|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}aq^{\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}-aq^{\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{\mathrm{i}\theta}},-{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}{\left(aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},-aq^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}};q^{\frac{1}{2}}\right)_{\infty}}\right|^{2}=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a^{2}q^{\frac{1}{2}}{\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,aq^{\frac{1}{4}})h(x,aq^{\frac{3}{4}})h(x,-aq^{\frac{1}{4}})h(x,-aq^{\frac{3}{4}})}}}}$ &

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

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

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:88

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:95 >>