Cunningham function
# 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

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

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:

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

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

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