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

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

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:

$x{X}^{″}+(x+1+m)X^{\prime }+(n+\frac{1}{2}m+1)X.$

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

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

References

• See exercise 10, chapter XVI, p. 353