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mathematical physics > special function > hypergeometric function > Cunningham function

mathematical analysis > functional analysis > special function > hypergeometric function > Cunningham function

probability theory > probability distribution > Cunningham function

mathematical statistics > probability distribution > Cunningham function

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Cunningham function

In statistics, the Cunningham function or Pearson–Cunningham function ω_m,n(x) is a generalisation of a special function introduced by Pearson (1906) and studied in the form here by Cunningham (1908). It can be defined in terms of the confluent hypergeometric function U, by

${\\displaystyle \\displaystyle \\omega _{m,n}(x)={\ rac {e^{-x+\\pi i(m/2-n)}}{\\Gamma (1+n-m/2)}}U(m/2-n,1+m,x).}$

The function was studied by Cunningham in the context of a multivariate generalisation of the Edgeworth expansion for approximating a probability density function based on its (joint) moments. In a more general context, the function is related to the solution of the constant-coefficient diffusion equation, in one or more dimensions.
The function ω_m,n(x) is a solution of the differential equation for X:

${\\displaystyle xX''+(x+1+m)X'+(n+{\ frac {1}{2}}m+1)X.}$

The special function studied by Pearson is given, in his notation by,

${\\displaystyle \\omega _{2n}(x)=\\omega _{0,n}(x).}$

Pearson-Cunningham function

fonction de Cunningham

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http://data.loterre.fr/ark:/67375/PSR-P7HZW9R6-S

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