In statistics, the Cunningham function or Pearson–Cunningham function ÃÂ<sub>m,n</sub>(x) is a generalisation of a special function introduced by and studied in the form here by . It can be defined in terms of the confluent hypergeometric function U, by
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 ÃÂ<sub>m,n</sub>(x) is a solution of the differential equation for X:
The special function studied by Pearson is given, in his notation by,