Kummer differential equation
= Kummer differential equation
{c}
Kummer's equation is
$$
xy''+(b-x)y'-ay=0.
$$
Its solution analytic at zero is the <confluent hypergeometric function of the first kind>
$$
M(a,b,x)=\sum_{m=0}^{\infty}
\frac{(a)_m}{(b)_m\,m!}x^m.
$$
When $b$ is not an integer, a second local solution is
$$
x^{1-b}M(a-b+1,2-b,x).
$$