Formula:KLS:14.05:48

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P n ⁡ ( c ; a , b , c , d ; q ) = q 1 2 ⁢ n ⁢ ( n + 1 ) ⁢ ( a ⁢ d c ) n ⁢ ( - q ⁢ b ⁢ c / d ; q ) n ( - q ⁢ a ⁢ d / c ; q ) n q-Jacobi-polynomial-four-parameters-P 𝑛 𝑐 𝑎 𝑏 𝑐 𝑑 𝑞 superscript 𝑞 1 2 𝑛 𝑛 1 superscript 𝑎 𝑑 𝑐 𝑛 q-Pochhammer-symbol 𝑞 𝑏 𝑐 𝑑 𝑞 𝑛 q-Pochhammer-symbol 𝑞 𝑎 𝑑 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(c;a,b,c,d;q\right)=q^{% \frac{1}{2}n(n+1)}\left(\frac{ad}{c}\right)^{n}\frac{\left(-qbc/d;q\right)_{n}% }{\left(-qad/c;q\right)_{n}}}}}

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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