Formula:KLS:14.05:28

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formula in Big q-Jacobi

Formula:KLS:14.05:29 >>

${\displaystyle{\displaystyle{\displaystyle\bigqJacobi{n}@{x}{a}{b}{0}{q}=\frac{\qPochhammer{bq}{q}{n}}{\qPochhammer{aq}{q}{n}}(-1)^{n}a^{n}q^{n+\binomial{n}{2}}\littleqJacobi{n}@{a^{-1}q^{-1}x}{b}{a}{q}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi

Bibliography

Equation in Section 14.5 of KLS.

URL links

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<< Formula:KLS:14.05:27

formula in Big q-Jacobi

Formula:KLS:14.05:29 >>