---
title: "Cunningham function"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Cunningham_function"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:22:16.014551+00:00"
instance: "kb-cron"
---

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 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:

  
    
      
        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, 

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


== Notes ==


== 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 {{citation}}: ISBN / Date incompatibility (help) See exercise 10, chapter XVI, p. 353