Formula:KLS:14.10:88

<< Formula:KLS:14.10:87

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:89 >>

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x|q\right)=\qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}}q^{\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q,-q^{\frac{1}{2}},-q}{q}{q}}}}$

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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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:14.10:87

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:89 >>