In statistics, the Cunningham function or Pearson - Cunningham function ωm,n(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
\displaystyle\omegam,n(x)=
| e-x+\pi | |
| \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 ωm,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,
\omega2n(x)=\omega0,n(x).