Formula:KLS:09.08:64

<< Formula:KLS:09.08:63

formula in Jacobi: Special cases

Formula:KLS:09.08:65 >>

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

Proof

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Symbols List

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

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:63

formula in Jacobi: Special cases

Formula:KLS:09.08:65 >>