Formula:KLS:14.10:110

<< Formula:KLS:14.10:109

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:111 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=q^{\frac{1}{4}n}\sum_{k=0}^{n}\frac{\left(q^{\frac{1}{2}};q\right)_{k}\left(q^{\frac{1}{2}};q\right)_{n-k}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.10 of KLS.

URL links

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<< Formula:KLS:14.10:109

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:111 >>