Confluent Hypergeometric Function - Kummer's Equation

Kummer's Equation

Kummer's equation is

,

with a regular singular point at 0 and an irregular singular point at ∞. It has two linearly independent solutions M(a,b,z) and U(a,b,z).

Kummer's function (of the first kind) M is a generalized hypergeometric series introduced in (Kummer 1837), given by

where

is the rising factorial. Another common notation for this solution is Φ(a,b,z). This defines an entire function of a.b, and z, except for poles at b = 0, −1, − 2, ...

Just as the confluent differential equation is a limit of the hypergeometric differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit of the hypergeometric function

and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function.

Another solution of Kummer's equation is the Tricomi confluent hypergeometric function U(a;b;z) introduced by Francesco Tricomi (1947), and sometimes denoted by Ψ(a;b;.z). The function U is defined in terms of Kummer's function M by

This is undefined for integer b, but can be extended to integer b by continuity.