Formula:KLS:14.12:17

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formula in Little q-Jacobi

Formula:KLS:14.12:18 >>

${\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)p_{n}\!\left(x;q^{\alpha},q^{\beta};q\right){}=\frac{q^{n\alpha+\binomial{n}{2}}(1-q)^{n}}{\left(q^{\alpha+1};q\right)_{n}}\left(\mathcal{D}_{q^{-1}}\right)^{n}\left[w(x;\alpha+n,\beta+n|q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;\alpha,\beta|q)=\frac{\left(qx;q\right)_{\infty}}{\left(q^{\beta+1}x;q\right)_{\infty}}x^{\alpha}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.12 of KLS.

URL links

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<< Formula:KLS:14.12:15

formula in Little q-Jacobi

Formula:KLS:14.12:18 >>