Formula:KLS:14.05:43

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lim n ( q a c - 1 x ) - n P n ( x ; a , b , c , d ; q ) = ( c x - 1 , - d x - 1 ; q ) ( - q a c - 1 d , q a ; q ) subscript 𝑛 superscript 𝑞 𝑎 superscript 𝑐 1 𝑥 𝑛 q-Jacobi-polynomial-four-parameters-P 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑞 q-Pochhammer-symbol 𝑐 superscript 𝑥 1 𝑑 superscript 𝑥 1 𝑞 q-Pochhammer-symbol 𝑞 𝑎 superscript 𝑐 1 𝑑 𝑞 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle\lim_{n\to\infty}(qac^{-1}x)^{-n}P_{% n}\!\left(x;a,b,c,d;q\right)=\frac{\left(cx^{-1},-dx^{-1};q\right)_{\infty}}{% \left(-qac^{-1}d,qa;q\right)_{\infty}}}}}

Proof

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Symbols List

P n subscript 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}}}}  : big q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial with four parameters : http://drmf.wmflabs.org/wiki/Definition:bigqJacobiIVparam
( a ; q ) n subscript 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}  : q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.5 of KLS.

URL links

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