kb/data/en.wikipedia.org/wiki/Cunningham_function-0.md

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