
1534 SPECIAL FUNCTIONS AND THEIR PROPERTIES
30.9.3 Whittaker Functions
The Whittaker functions M
k,µ
(x) and W
k,µ
(x) are linearly independent solutions of the
Whittaker equation:
y
′′
xx
+
−
1
4
+
1
2
k +
1
4
− µ
2
x
−2
y = 0.
The Whittaker functions are expressed in terms of degenerate hypergeometric functions as
M
k,µ
(x) = x
µ+1/2
e
−x/2
Φ
1
2
+ µ − k, 1 + 2µ; x
,
W
k,µ
(x) = x
µ+1/2
e
−x/2
Ψ
1
2
+ µ − k, 1 + 2µ; x
.
30.10 Hypergeometric Functions
30.10.1 Various Representations o f the Hypergeometric Function
◮ Representations of the hypergeometric function via hypergeometric series.
The hypergeometric function F (α, β, γ; x) is a solution of the Gaussian hypergeometric
equation ...