## 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).