Formula:KLS:14.12:15

<< Formula:KLS:14.12:14

formula in Little q-Jacobi

Formula:KLS:14.12:17 >>

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

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
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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:14.12:14

formula in Little q-Jacobi

Formula:KLS:14.12:17 >>