Formula:KLS:09.08:66

<< Formula:KLS:09.08:65

formula in Jacobi: Special cases

Formula:KLS:09.08:67 >>

${\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}}{}=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(-\gamma\right)_{n}}}{{\left(\frac{1}{2}\right)_{n}}n!}T_{n}\left(x\right)t^{n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle T_{n}}}}$ : Chebyshev polynomial of the first kind : http://dlmf.nist.gov/18.3#T1.t1.r8

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:65

formula in Jacobi: Special cases

Formula:KLS:09.08:67 >>