114
8 Hypergeometric Functions
-10
-5
0
5
10
-2
0
2
4
2F1
-10
-5
0
5
10
-10
0
10
-10
-5
0
5
10
x
-2
-1
0
1
2
2F1
-10
-5
0
5
10
x
-2
0
2
Fig. 8.1 The hypergeometric function 2 F 1 (a, b; c; x). Solid lines the real function values, and
dotted lines the imaginary function values. b = 1.25, top left: a = 0.75, c = 2.5, top right
a = 2 · 0.75, c = 2.5, bottom left: a = 0.75, c = 2 · 2.5, bottom right a = 2 · 0.75, c = 2 · 2.5
is a solution of the hypergeometric or Gauss differential equation
z(z − 1)
d 2 F
dz 2 = a · b · F − [c − (a + b + 1)z]
dF
dz
.
(8.2)
The hypergeometric series 2 F 1 (b, a; c; z) is absolutely convergent for |z| < 1,
divergent for |z| > 0, and for (c − a − b) > 0 convergent on the boundary |z| = 1.
With the help of several linear transformations, the hypergeometric function has an
analytic continuation outside the unit circle; some examples are plotted in Fig. 8.1.
Transformation Formulae
The hypergeometric series, Eq. (8.1), can be continued for |z| > 1 via
2 F 1 (a, b; c; z) = 2 F 1 (b, a; c; z)
(8.3)
= (1 − z) c−a−b
2 F 1 (c − a, c − b; c; z)
(8.4)
= (1 − z) −a
2 F 1 (a, c − b; c;
z
z − 1
)
(8.5)
= (1 − z) −b
2 F 1 (c − a, b; c;
z
z − 1
)
(8.6)
2 F 1 (a, b; c; z) =
Γ (c)Γ (c − a − b)
Γ (c − a)Γ (c − b)
2 F 1 (a, b; a + b − c + 1; 1 − z)
(8.7)
+
Γ (c)Γ (a + b − c)
Γ (a)Γ (b)
(1 − z) c−a−b
× 2 F 1 (c − a, c − b; c − a − b + 1; 1 − z)