Formula:KLS:14.05:28

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P n ( x ; a , b , 0 ; q ) = ( b q ; q ) n ( a q ; q ) n ( - 1 ) n a n q n + \binomial n 2 p n ( a - 1 q - 1 x ; b , a ; q ) big-q-Jacobi-polynomial-P 𝑛 𝑥 𝑎 𝑏 0 𝑞 q-Pochhammer-symbol 𝑏 𝑞 𝑞 𝑛 q-Pochhammer-symbol 𝑎 𝑞 𝑞 𝑛 superscript 1 𝑛 superscript 𝑎 𝑛 superscript 𝑞 𝑛 \binomial 𝑛 2 little-q-Jacobi-polynomial-p 𝑛 superscript 𝑎 1 superscript 𝑞 1 𝑥 𝑏 𝑎 𝑞 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;a,b,0;q\right)=\frac% {\left(bq;q\right)_{n}}{\left(aq;q\right)_{n}}(-1)^{n}a^{n}q^{n+\binomial{n}{2% }}p_{n}\!\left(a^{-1}q^{-1}x;b,a;q\right)}}}

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 : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi
( 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
( n k ) binomial 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi

Bibliography

Equation in Section 14.5 of KLS.

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