Formula:KLS:14.05:17

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formula in Big q-Jacobi

Formula:KLS:14.05:18 >>

${\displaystyle{\displaystyle{\displaystyle w(x;a,b,c;q)P_{n}\!\left(x;a,b,c;q\right){}=\frac{a^{n}c^{n}q^{n(n+1)}(1-q)^{n}}{\left(aq,cq;q\right)_{n}}\left(\mathcal{D}_{q}\right)^{n}\left[w(x;aq^{n},bq^{n},cq^{n};q)\right]}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;a,b,c;q)=\frac{\left(a^{-1}x,c^{-1}x;q\right)_{\infty}}{\left(x,bc^{-1}x;q\right)_{\infty}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}^{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-derivative : http://drmf.wmflabs.org/wiki/Definition:qderiv

Bibliography

Equation in Section 14.5 of KLS.

URL links

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formula in Big q-Jacobi

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