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

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

The function was studied by Cunningham¹ 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 ω_m,n(x) is a solution of the differential equation for X:¹

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

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

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