Formula:KLS:09.03:23

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formula in Continuous dual Hahn

Formula:KLS:09.03:24 >>

${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{2}{2}@@{a+\mathrm{i}x,a-\mathrm{i}x}{a+b,a+c}{-t}=\sum_{n=0}^{\infty}\frac{S_{n}\!\left(x^{2};a,b,c\right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}n!}t^{n}}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle S_{n}}}}$ : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.3 of KLS.

URL links

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<< Formula:KLS:09.03:22

formula in Continuous dual Hahn

Formula:KLS:09.03:24 >>