Formula:KLS:14.10:105

<< Formula:KLS:14.10:104

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:106 >>

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{\frac{1}{4}}{\mathrm{e}^{\mathrm{i}\theta}},q^{\frac{3}{4}}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}\ \qHyperrphis{2}{1}@@{-q^{\frac{1}{4}}{\mathrm{e}^{-\mathrm{i}\theta}},-q^{\frac{3}{4}}{\mathrm{e}^{-\mathrm{i}\theta}}}{q}{q}{{\mathrm{e}^{\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(-q^{\frac{1}{2}},-q;q\right)_{n}}{\left(q,q;q\right)_{n}}\frac{P_{n}\!\left(x|q\right)}{q^{\frac{1}{4}n}}t^{n}}}}$

Substitution(s)

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

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : continuous ${\displaystyle{\displaystyle{\displaystyle q}}}$-Legendre polynomial : http://drmf.wmflabs.org/wiki/Definition:ctsqLegendre
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : 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:14.10:104

formula in Continuous q-Jacobi: Special cases

Formula:KLS:14.10:106 >>