Formula:KLS:09.03:03: Difference between revisions

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Latest revision as of 06:34, 22 December 2019


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_0^{\infty}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 \ctsdualHahn{m}@{x^2}{a}{b}{c}\ctsdualHahn{n}@{x^2}{a}{b}{c}\,dx {}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{b-a}\EulerGamma@{c-a}}{\EulerGamma@{-2a}} {}\sum_{\begin{array}{c} {\scriptstyle k=0,1,2\ldots}\\ {\scriptstyle a+k<0}\end{array}} \frac{\pochhammer{2a}{k}\pochhammer{a+1}{k}\pochhammer{a+b}{k}\pochhammer{a+c}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}k!}(-1)^k {} \ctsdualHahn{m}@{-(a+k)^2}{a}{b}{c}\ctsdualHahn{n}@{-(a+k)^2}{a}{b}{c} {}=\EulerGamma@{n+a+b}\EulerGamma@{n+a+c}\EulerGamma@{n+b+c}n!\,\Kronecker{m}{n} }}

Proof

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Symbols List

 : integral : http://dlmf.nist.gov/1.4#iv
 : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
 : imaginary unit : http://dlmf.nist.gov/1.9.i
 : continuous dual Hahn polynomial : http://dlmf.nist.gov/18.25#T1.t1.r3
 : sum : http://drmf.wmflabs.org/wiki/Definition:sum
 : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
 : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.3 of KLS.

URL links

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