Formula:KLS:14.12:26

<< Formula:KLS:14.12:25

formula in Little q-Jacobi

Little q-Jacobi >>

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b;q\right)=\frac{\left(bq;q\right)_{n}}{\left(aq;q\right)_{n}}(-1)^{n}b^{-n}q^{-n-\binomial{n}{2}}P_{n}\!\left(bqx;b,a,0;q\right)}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle p_{n}}}}$ : little ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqJacobi

Bibliography

Equation in Section 14.12 of KLS.

URL links

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<< Formula:KLS:14.12:25

formula in Little q-Jacobi

Little q-Jacobi >>