Cunningham function

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In statistics, the Cunningham function or Pearson–Cunningham function ωm,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

The function was studied by Cunningham[1] 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.[1]

The function ωm,n(x) is a solution of the differential equation for X:[1]

The special function studied by Pearson is given, in his notation by,[1]