# Cunningham function

Updated on

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

ω m , n ( x ) = e x + π i ( m / 2 n ) Γ ( 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 ωm,n(x) is a solution of the differential equation for X:

x X + ( x + 1 + m ) X + ( n + 1 2 m + 1 ) X .

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

ω 2 n ( x ) = ω 0 , n ( x ) .

## References

Cunningham function Wikipedia

Topics