In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell (1880) and that generalize Gauss's hypergeometric series 2F1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable.
The Appell series F1 is defined for |x| < 1, |y| < 1 by the double series:
F
1
(
a
,
b
1
,
b
2
;
c
;
x
,
y
)
=
∑
m
,
n
=
0
∞
(
a
)
m
+
n
(
b
1
)
m
(
b
2
)
n
(
c
)
m
+
n
m
!
n
!
x
m
y
n
,
where the Pochhammer symbol (q)n represents the rising factorial:
(
q
)
n
=
q
(
q
+
1
)
⋯
(
q
+
n
−
1
)
=
Γ
(
q
+
n
)
Γ
(
q
)
,
where the second equality is true for all complex
q
except
q
=
0
,
−
1
,
−
2
,
…
.
For other values of x and y the function F1 can be defined by analytic continuation.
Similarly, the function F2 is defined for |x| + |y| < 1 by the series:
F
2
(
a
,
b
1
,
b
2
;
c
1
,
c
2
;
x
,
y
)
=
∑
m
,
n
=
0
∞
(
a
)
m
+
n
(
b
1
)
m
(
b
2
)
n
(
c
1
)
m
(
c
2
)
n
m
!
n
!
x
m
y
n
,
the function F3 for |x| < 1, |y| < 1 by the series:
F
3
(
a
1
,
a
2
,
b
1
,
b
2
;
c
;
x
,
y
)
=
∑
m
,
n
=
0
∞
(
a
1
)
m
(
a
2
)
n
(
b
1
)
m
(
b
2
)
n
(
c
)
m
+
n
m
!
n
!
x
m
y
n
,
and the function F4 for |x|½ + |y|½ < 1 by the series:
F
4
(
a
,
b
;
c
1
,
c
2
;
x
,
y
)
=
∑
m
,
n
=
0
∞
(
a
)
m
+
n
(
b
)
m
+
n
(
c
1
)
m
(
c
2
)
n
m
!
n
!
x
m
y
n
.
Like the Gauss hypergeometric series 2F1, the Appell double series entail recurrence relations among contiguous functions. For example, a basic set of such relations for Appell's F1 is given by:
(
a
−
b
1
−
b
2
)
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
−
a
F
1
(
a
+
1
,
b
1
,
b
2
,
c
;
x
,
y
)
+
b
1
F
1
(
a
,
b
1
+
1
,
b
2
,
c
;
x
,
y
)
+
b
2
F
1
(
a
,
b
1
,
b
2
+
1
,
c
;
x
,
y
)
=
0
,
c
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
−
(
c
−
a
)
F
1
(
a
,
b
1
,
b
2
,
c
+
1
;
x
,
y
)
−
a
F
1
(
a
+
1
,
b
1
,
b
2
,
c
+
1
;
x
,
y
)
=
0
,
c
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
+
c
(
x
−
1
)
F
1
(
a
,
b
1
+
1
,
b
2
,
c
;
x
,
y
)
−
(
c
−
a
)
x
F
1
(
a
,
b
1
+
1
,
b
2
,
c
+
1
;
x
,
y
)
=
0
,
c
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
+
c
(
y
−
1
)
F
1
(
a
,
b
1
,
b
2
+
1
,
c
;
x
,
y
)
−
(
c
−
a
)
y
F
1
(
a
,
b
1
,
b
2
+
1
,
c
+
1
;
x
,
y
)
=
0
.
Any other relation valid for F1 can be derived from these four.
Similarly, all recurrence relations for Appell's F3 follow from this set of five:
c
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
+
(
a
1
+
a
2
−
c
)
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
+
1
;
x
,
y
)
−
a
1
F
3
(
a
1
+
1
,
a
2
,
b
1
,
b
2
,
c
+
1
;
x
,
y
)
−
a
2
F
3
(
a
1
,
a
2
+
1
,
b
1
,
b
2
,
c
+
1
;
x
,
y
)
=
0
,
c
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
−
c
F
3
(
a
1
+
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
+
b
1
x
F
3
(
a
1
+
1
,
a
2
,
b
1
+
1
,
b
2
,
c
+
1
;
x
,
y
)
=
0
,
c
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
−
c
F
3
(
a
1
,
a
2
+
1
,
b
1
,
b
2
,
c
;
x
,
y
)
+
b
2
y
F
3
(
a
1
,
a
2
+
1
,
b
1
,
b
2
+
1
,
c
+
1
;
x
,
y
)
=
0
,
c
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
−
c
F
3
(
a
1
,
a
2
,
b
1
+
1
,
b
2
,
c
;
x
,
y
)
+
a
1
x
F
3
(
a
1
+
1
,
a
2
,
b
1
+
1
,
b
2
,
c
+
1
;
x
,
y
)
=
0
,
c
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
−
c
F
3
(
a
1
,
a
2
,
b
1
,
b
2
+
1
,
c
;
x
,
y
)
+
a
2
y
F
3
(
a
1
,
a
2
+
1
,
b
1
,
b
2
+
1
,
c
+
1
;
x
,
y
)
=
0
.
Derivatives and differential equations
For Appell's F1, the following derivatives result from the definition by a double series:
∂
n
∂
x
n
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
=
(
a
)
n
(
b
1
)
n
(
c
)
n
F
1
(
a
+
n
,
b
1
+
n
,
b
2
,
c
+
n
;
x
,
y
)
∂
n
∂
y
n
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
=
(
a
)
n
(
b
2
)
n
(
c
)
n
F
1
(
a
+
n
,
b
1
,
b
2
+
n
,
c
+
n
;
x
,
y
)
From its definition, Appell's F1 is further found to satisfy the following system of second-order differential equations:
x
(
1
−
x
)
∂
2
F
1
(
x
,
y
)
∂
x
2
+
y
(
1
−
x
)
∂
2
F
1
(
x
,
y
)
∂
x
∂
y
+
[
c
−
(
a
+
b
1
+
1
)
x
]
∂
F
1
(
x
,
y
)
∂
x
−
b
1
y
∂
F
1
(
x
,
y
)
∂
y
−
a
b
1
F
1
(
x
,
y
)
=
0
y
(
1
−
y
)
∂
2
F
1
(
x
,
y
)
∂
y
2
+
x
(
1
−
y
)
∂
2
F
1
(
x
,
y
)
∂
x
∂
y
+
[
c
−
(
a
+
b
2
+
1
)
y
]
∂
F
1
(
x
,
y
)
∂
y
−
b
2
x
∂
F
1
(
x
,
y
)
∂
x
−
a
b
2
F
1
(
x
,
y
)
=
0
A system partial differential equations for F2 is
x
(
1
−
x
)
∂
2
F
2
(
x
,
y
)
∂
x
2
−
x
y
∂
2
F
2
(
x
,
y
)
∂
x
∂
y
+
[
c
1
−
(
a
+
b
1
+
1
)
x
]
∂
F
2
(
x
,
y
)
∂
x
−
b
1
y
∂
F
2
(
x
,
y
)
∂
y
−
a
b
1
F
2
(
x
,
y
)
=
0
y
(
1
−
y
)
∂
2
F
2
(
x
,
y
)
∂
y
2
−
x
y
∂
2
F
2
(
x
,
y
)
∂
x
∂
y
+
[
c
2
−
(
a
+
b
2
+
1
)
x
]
∂
F
2
(
x
,
y
)
∂
y
−
b
2
x
∂
F
2
(
x
,
y
)
∂
x
−
a
b
2
F
2
(
x
,
y
)
=
0
The system have solution
F
2
(
x
,
y
)
=
C
1
F
2
(
a
,
b
1
,
b
2
,
c
1
,
c
2
;
x
,
y
)
+
C
2
x
1
−
c
1
F
2
(
a
−
c
1
+
1
,
b
1
−
c
1
+
1
,
b
2
,
2
−
c
1
,
c
2
;
x
,
y
)
+
C
3
y
1
−
c
2
F
2
(
a
−
c
2
+
1
,
b
1
,
b
2
−
c
2
+
1
,
c
1
,
2
−
c
2
;
x
,
y
)
+
C
4
x
1
−
c
1
y
1
−
c
2
F
2
(
a
−
c
1
−
c
2
+
2
,
b
1
−
c
1
+
1
,
b
2
−
c
2
+
1
,
2
−
c
1
,
2
−
c
2
;
x
,
y
)
Similarly, for F3 the following derivatives result from the definition:
∂
∂
x
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
=
a
1
b
1
c
F
3
(
a
1
+
1
,
a
2
,
b
1
+
1
,
b
2
,
c
+
1
;
x
,
y
)
∂
∂
y
F
3
(
a
1
,
a
2
,
b
1
,
b
2
,
c
;
x
,
y
)
=
a
2
b
2
c
F
3
(
a
1
,
a
2
+
1
,
b
1
,
b
2
+
1
,
c
+
1
;
x
,
y
)
And for F3 the following system of differential equations is obtained:
x
(
1
−
x
)
∂
2
F
3
(
x
,
y
)
∂
x
2
+
y
∂
2
F
3
(
x
,
y
)
∂
x
∂
y
+
[
c
−
(
a
1
+
b
1
+
1
)
x
]
∂
F
3
(
x
,
y
)
∂
x
−
a
1
b
1
F
3
(
x
,
y
)
=
0
y
(
1
−
y
)
∂
2
F
3
(
x
,
y
)
∂
y
2
+
x
∂
2
F
3
(
x
,
y
)
∂
x
∂
y
+
[
c
−
(
a
2
+
b
2
+
1
)
y
]
∂
F
3
(
x
,
y
)
∂
y
−
a
2
b
2
F
3
(
x
,
y
)
=
0
A system partial differential equations for F4 is
x
(
1
−
x
)
∂
2
F
4
(
x
,
y
)
∂
x
2
−
y
2
∂
2
F
4
(
x
,
y
)
∂
y
2
−
2
x
y
∂
2
F
4
(
x
,
y
)
∂
x
∂
y
+
[
c
1
−
(
a
+
b
+
1
)
x
]
∂
F
4
(
x
,
y
)
∂
x
−
(
a
+
b
+
1
)
y
∂
F
4
(
x
,
y
)
∂
y
−
a
b
F
4
(
x
,
y
)
=
0
y
(
1
−
y
)
∂
2
F
4
(
x
,
y
)
∂
y
2
−
x
2
∂
2
F
4
(
x
,
y
)
∂
x
2
−
2
x
y
∂
2
F
4
(
x
,
y
)
∂
x
∂
y
+
[
c
2
−
(
a
+
b
+
1
)
y
]
∂
F
4
(
x
,
y
)
∂
y
−
(
a
+
b
+
1
)
x
∂
F
4
(
x
,
y
)
∂
x
−
a
b
F
4
(
x
,
y
)
=
0
The system have solution
F
4
(
x
,
y
)
=
C
1
F
4
(
a
,
b
,
c
1
,
c
2
;
x
,
y
)
+
C
2
x
1
−
c
1
F
4
(
a
−
c
1
+
1
,
b
−
c
1
+
1
,
2
−
c
1
,
c
2
;
x
,
y
)
+
C
3
y
1
−
c
1
F
4
(
a
−
c
2
+
1
,
b
−
c
2
+
1
,
c
1
,
2
−
c
2
;
x
,
y
)
+
C
4
x
1
−
c
1
y
1
−
c
2
F
4
(
2
+
a
−
c
1
+
c
2
,
2
+
b
−
c
1
+
c
2
,
2
−
c
1
,
2
−
c
2
;
x
,
y
)
The four functions defined by Appell's double series can be represented in terms of double integrals involving elementary functions only (Gradshteyn & Ryzhik 2015, §9.184). However, Émile Picard (1881) discovered that Appell's F1 can also be written as a one-dimensional Euler-type integral:
F
1
(
a
,
b
1
,
b
2
,
c
;
x
,
y
)
=
Γ
(
c
)
Γ
(
a
)
Γ
(
c
−
a
)
∫
0
1
t
a
−
1
(
1
−
t
)
c
−
a
−
1
(
1
−
x
t
)
−
b
1
(
1
−
y
t
)
−
b
2
d
t
,
ℜ
c
>
ℜ
a
>
0
.
This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration.
Picard's integral representation implies that the incomplete elliptic integrals F and E as well as the complete elliptic integral Π are special cases of Appell's F1:
F
(
ϕ
,
k
)
=
∫
0
ϕ
d
θ
1
−
k
2
sin
2
θ
=
sin
(
ϕ
)
F
1
(
1
2
,
1
2
,
1
2
,
3
2
;
sin
2
ϕ
,
k
2
sin
2
ϕ
)
,
|
ℜ
ϕ
|
<
π
2
,
E
(
ϕ
,
k
)
=
∫
0
ϕ
1
−
k
2
sin
2
θ
d
θ
=
sin
(
ϕ
)
F
1
(
1
2
,
1
2
,
−
1
2
,
3
2
;
sin
2
ϕ
,
k
2
sin
2
ϕ
)
,
|
ℜ
ϕ
|
<
π
2
,
Π
(
n
,
k
)
=
∫
0
π
/
2
d
θ
(
1
−
n
sin
2
θ
)
1
−
k
2
sin
2
θ
=
π
2
F
1
(
1
2
,
1
,
1
2
,
1
;
n
,
k
2
)
.
There are seven related series of two variables, Φ
1, Φ
2, Φ
3, Ψ
1, Ψ
2, Ξ
1, and Ξ
2, which generalize Kummer's confluent hypergeometric function
1F1 of one variable and the confluent hypergeometric limit function
0F1 of one variable in a similar manner. The first of these was introduced by Pierre Humbert in 1920.
Giuseppe Lauricella (1893) defined four functions similar to the Appell series, but depending on many variables rather than just the two variables
x and
y. These series were also studied by Appell. They satisfy certain partial differential equations, and can also be given in terms of Euler-type integrals and contour integrals.
