Cunningham function

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 ) . {\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 Cunningham[1] 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 ωm,n(x) is a solution of the differential equation for X:[1]

x X ″ + ( x + 1 + m ) X ′ + ( n + 1 2 m + 1 ) X . {\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]

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

Notes

  1. ^ a b c d Cunningham (1908)

References

  • Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 13". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 510. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  • Cunningham, E. (1908), "The ω-Functions, a Class of Normal Functions Occurring in Statistics", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 81 (548), The Royal Society: 310–331, doi:10.1098/rspa.1908.0085, ISSN 0950-1207, JSTOR 93061
  • Pearson, Karl (1906), A mathematical theory of random migration, London, Dulau and co.
  • Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2 See exercise 10, chapter XVI, p. 353


  • v
  • t
  • e