Formula:KLS:14.05:76

<< Formula:KLS:14.05:72

formula in Big q-Jacobi: Special case

Formula:KLS:14.05:77 >>

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : big ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:bigqLegendre
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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:KLS:14.05:72

formula in Big q-Jacobi: Special case

Formula:KLS:14.05:77 >>