Formula:KLS:09.07:14

<< Formula:KLS:09.07:13

formula in Meixner-Pollaczek

Formula:KLS:09.07:15 >>

${\displaystyle{\displaystyle{\displaystyle\expe^{t}\,{{}_{1}F_{1}}\!\left({\lambda+\iunit x\atop 2\lambda};(\expe^{-2\iunit\phi}-1)t\right)=\sum_{n=0}^{\infty}\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{\pochhammer{2\lambda}{n}\expe^{\iunit n\phi}}t^{n}}}}$

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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}$ : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii

Bibliography

Equation in Section 9.7 of KLS.

URL links

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<< Formula:KLS:09.07:13

formula in Meixner-Pollaczek

Formula:KLS:09.07:15 >>