Formula:KLS:14.05:44

<< Formula:KLS:14.05:43

formula in Big q-Jacobi

Formula:KLS:14.05:45 >>

${\displaystyle{\displaystyle{\displaystyle\lim_{n\to\infty}(qac^{-1}x)^{-n}P_{n}\!\left(x;a,b,c,d;q\right)=\frac{\left(qb,cx^{-1};q\right)_{n}}{\left(qa,-qac^{-1}d;q\right)_{n}}\qHyperrphis{1}{1}@@{-qbd^{-1}x}{qb}{q}{-dx^{-1}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:43

formula in Big q-Jacobi

Formula:KLS:14.05:45 >>