Formula:KLS:09.08:95

<< Formula:KLS:09.08:94

formula in Jacobi: Special cases

Formula:KLS:09.08:96 >>

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@{-}{1}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{\LegendrePoly{n}@{x}}{n!}t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}$ : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle P_{n}}}}$ : Legendre polynomial : http://dlmf.nist.gov/18.3#T1.t1.r25

Bibliography

Equation in Section 9.8 of KLS.

URL links

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<< Formula:KLS:09.08:94

formula in Jacobi: Special cases

Formula:KLS:09.08:96 >>