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
\( \displaystyle \omega _{{m,n}}(x)={\frac {e^{{-x+\pi i(m/2-n)}}}{\Gamma (1+n-m/2)}}U(m/2-n,1+m,x). \)
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]
\( xX''+(x+1+m)X'+(n+{\tfrac {1}{2}}m+1)X. \)
The special function studied by Pearson is given, in his notation by,[1]
\) \omega _{{2n}}(x)=\omega _{{0,n}}(x). \)
Notes
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. 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, The Royal Society, 81 (548): 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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