Formula:KLS:14.12:07: Difference between revisions

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Latest revision as of 08:37, 22 December 2019


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

Proof

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Symbols List

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
( 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
( 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
p ^ n subscript ^ 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{p}}_{n}}}}  : monic little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:moniclittleqJacobi

Bibliography

Equation in Section 14.12 of KLS.

URL links

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