In statistics, the **Cunningham function** or **Pearson–Cunningham function** ω_{m,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
)
.
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 ω_{m,n}(*x*) is a solution of the differential equation for *X*:

x
X
″
+
(
x
+
1
+
m
)
X
′
+
(
n
+
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
)
.