Formula:KLS:09.08:15

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formula in Jacobi: Special cases

Formula:KLS:09.08:16 >>

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{xt}}\,\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x^{2}-1)t^{2}}{4}}=\sum_{n=0}^{\infty}\frac{C^{\lambda}_{n}\left(x\right)}{{\left(2\lambda\right)_{n}}}t^{n}}}}$

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
${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : 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:09.08:14

formula in Jacobi: Special cases

Formula:KLS:09.08:16 >>