In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function or just Wright function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on an idea of E. Maitland Wright (1935):
p
Ψ
q
[
(
a
1
,
A
1
)
(
a
2
,
A
2
)
…
(
a
p
,
A
p
)
(
b
1
,
B
1
)
(
b
2
,
B
2
)
…
(
b
q
,
B
q
)
;
z
]
=
∑
n
=
0
∞
Γ
(
a
1
+
A
1
n
)
⋯
Γ
(
a
p
+
A
p
n
)
Γ
(
b
1
+
B
1
n
)
⋯
Γ
(
b
q
+
B
q
n
)
z
n
n
!
.
Its normalisation
p
Ψ
q
∗
[
(
a
1
,
A
1
)
(
a
2
,
A
2
)
…
(
a
p
,
A
p
)
(
b
1
,
B
1
)
(
b
2
,
B
2
)
…
(
b
q
,
B
q
)
;
z
]
=
Γ
(
b
1
)
⋯
Γ
(
b
q
)
Γ
(
a
1
)
⋯
Γ
(
a
p
)
∑
n
=
0
∞
Γ
(
a
1
+
A
1
n
)
⋯
Γ
(
a
p
+
A
p
n
)
Γ
(
b
1
+
B
1
n
)
⋯
Γ
(
b
q
+
B
q
n
)
z
n
n
!
becomes pFq(z) for A1...p = B1...q = 1.
The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50):
p
Ψ
q
[
(
a
1
,
A
1
)
(
a
2
,
A
2
)
…
(
a
p
,
A
p
)
(
b
1
,
B
1
)
(
b
2
,
B
2
)
…
(
b
q
,
B
q
)
;
z
]
=
H
p
,
q
+
1
1
,
p
[
−
z
|
(
1
−
a
1
,
A
1
)
(
1
−
a
2
,
A
2
)
…
(
1
−
a
p
,
A
p
)
(
0
,
1
)
(
1
−
b
1
,
B
1
)
(
1
−
b
2
,
B
2
)
…
(
1
−
b
q
,
B
q
)
]
.
