# Formula:KLS:14.05:58

$\displaystyle {\displaystyle \int_{z_- q^\mathbb{Z}\cup z_+ q^\mathbb{Z}}\bigqJacobi{m}@{cx}{c/b}{d/a}{c/a}{q} \bigqJacobi{n}@{cx}{c/b}{d/a}{c/a}{q} \frac{\qPochhammer{ax,bx}{q}{\infty}}{\qPochhammer{cx,dx}{q}{\infty}} d_qx=h_n\Kronecker{m}{n} (m,n=0,1,\ldots,N) }$

## Substitution(s)

$\displaystyle {\displaystyle h_0=(1-q)c \frac{\qPochhammer{q,-d/c,-qc/d,q^2ab}{q}{\infty}} {\qPochhammer{qa,qb,-qbc/d,-qad/c}{q}{\infty}}}$ &
$\displaystyle {\displaystyle h_0 =(1-q)z_+ \frac{\qPochhammer{q,a/c,a/d,b/c,b/d}{q}{\infty}}{\qPochhammer{ab/(qcd)}{q}{\infty}} \frac{\theta(z_-/z_+,cdz_-z_+;q)}{\theta(cz_-,dz_-,cz_+,dz_+;q)}}$

## Proof

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## Symbols List

& : logical and
$\displaystyle {\displaystyle \int}$  : integral : http://dlmf.nist.gov/1.4#iv
$\displaystyle {\displaystyle P_{n}}$  : big $\displaystyle {\displaystyle q}$ -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi
$\displaystyle {\displaystyle (a;q)_n}$  : $\displaystyle {\displaystyle q}$ -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
$\displaystyle {\displaystyle \delta_{m,n}}$  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4