Formula:KLS:14.05:37

<< Formula:KLS:14.05:36

formula in Big q-Jacobi

Formula:KLS:14.05:38 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-d}^{c}P_{m}\!\left(x;a,b,c,d;q\right)P_{n}\!\left(x;a,b,c,d;q\right)\frac{\left(qx/c,-qx/d;q\right)_{\infty}}{\left(qax/c,-qbx/d;q\right)_{\infty}}d_{q}x=h_{n}\delta_{m,n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)c\frac{\left(q,-d/c,-qc/d,q^{2}ab;q\right)_{\infty}}{\left(qa,qb,-qbc/d,-qad/c;q\right)_{\infty}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{0}=(1-q)z_{+}\frac{\left(q,a/c,a/d,b/c,b/d;q\right)_{\infty}}{\left(ab/(qcd);q\right)_{\infty}}\frac{\theta(z_{-}/z_{+},cdz_{-}z_{+};q)}{\theta(cz_{-},dz_{-},cz_{+},dz_{+};q)}}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial with four parameters : http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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

Bibliography

Equation in Section 14.5 of KLS.

URL links

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formula in Big q-Jacobi

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