Formula:KLS:14.05:66

<< Formula:KLS:14.05:65

formula in Big q-Jacobi: Special case

Formula:KLS:14.05:67 >>

${\displaystyle{\displaystyle{\displaystyle\int_{cq}^{q}P_{m}\!\left(x;c;q\right)P_{n}\!\left(x;c;q\right)\,d_{q}x{}=q(1-c)\frac{(1-q)}{(1-q^{2n+1})}\frac{\left(c^{-1}q;q\right)_{n}}{\left(cq;q\right)_{n}}(-cq^{2})^{n}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle c<0}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqLegendre
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:65

formula in Big q-Jacobi: Special case

Formula:KLS:14.05:67 >>