ABSTRACT
The confluent hypergeometric function, denoted by 1F1(a; c; z), is defined by
(13.1.1) 1F1(a; c; z) =
(a)nz n
(c)nn!
with (a)n being the rising factorial
(a)n := a(a+ 1) · · · (a+ n− 1) = Γ(a+ n) Γ(a)
,
for a ∈ C. It arises when two of the regular singular points of the differential equation for the Gauss hypergeometric function 2F1(a, b; c; z), given by
(13.1.2) z(1− z)y′′ + (c− (a+ b+ 1)z)y′ − aby = 0, are allowed to merge into one singular point. More specifically, if we replace z by z/b in 2F1(a, b; c; z), then the corresponding differential equation has singular points at 0, b, and ∞. Now let b → ∞ so as to have infinity as a confluence of two singularities. This results in the function 1F1(a; c; z) so that
(13.1.3) 1F1(a; c; z) = lim b→∞ 2
F1
( a, b; c;
z
b
) ,
and the corresponding differential equation
(13.1.4) zy′′ + (c− z)y′ − ay = 0, known as the confluent hypergeometric equation. The following two transformation formulas for 1F1, due to Kummer, are very useful:
1F1(a; c; z) = e z 1F1(c− a; c;−z) (b 6= 0,−1,−2, · · · ),
1F1(a; 2a; 2z) = e z 0F1
( −; a+ 1
2 ; z2
) (2a is not an odd integer < 0).