Confluent hypergeometric functions

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. (The term "confluent" refers to the merging of singular points of families of differential equations; "confluere" is Latin for "to flow together".) There are several common standard forms of confluent hypergeometric functions:

Kummer's equation may be written as:

with a regular singular point at ${\displaystyle z=0}$ and an irregular singular point at ${\displaystyle z=\infty }$. It has two (usually) 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:

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