176 lines
4.0 KiB
Markdown
176 lines
4.0 KiB
Markdown
---
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title: "Cunningham function"
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chunk: 1/1
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source: "https://en.wikipedia.org/wiki/Cunningham_function"
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category: "reference"
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tags: "science, encyclopedia"
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date_saved: "2026-05-05T12:22:16.014551+00:00"
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instance: "kb-cron"
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---
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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
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{\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).}
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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.
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The function ωm,n(x) is a solution of the differential equation for X:
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{\displaystyle xX''+(x+1+m)X'+(n+{\tfrac {1}{2}}m+1)X.}
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The special function studied by Pearson is given, in his notation by,
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ω
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{\displaystyle \omega _{2n}(x)=\omega _{0,n}(x).}
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== Notes ==
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== References ==
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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.
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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
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Pearson, Karl (1906), A mathematical theory of random migration, London, Dulau and co.
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Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2 {{citation}}: ISBN / Date incompatibility (help) See exercise 10, chapter XVI, p. 353 |