Confluent hypergeometric function

The term confluent refers to the merging of singular points of families of differential equations; confluere is Latin for "to flow together".

Some values of a and b yield solutions that can be expressed in terms of other known functions.

When a is a non-positive integer, then Kummer's function (if it is defined) is a generalized Laguerre polynomial.

For most combinations of real or complex a and b, the functions M(a, b, z) and U(a, b, z) are independent, and if b is a non-positive integer, so M(a, b, z) doesn't exist, then we may be able to use z1−bM(a+1−b, 2−b, z) as a second solution.

Thus Confluent Hypergeometric Functions can be used to solve "most" second-order ordinary differential equations whose variable coefficients are all linear functions of z, because they can be transformed to the Extended Confluent Hypergeometric Equation.

If Re b > Re a > 0, M(a, b, z) can be represented as an integral thus M(a, a+b, it) is the characteristic function of the beta distribution.

If z = x ∈ R, then making a change of variables in the integral followed by expanding the binomial series and integrating it formally term by term gives rise to an asymptotic series expansion, valid as x → ∞:[2] where

The asymptotic behavior of Kummer's solution for large |z| is: The powers of z are taken using −3π/2 < arg z ≤ π/2.

[3] The first term is not needed when Γ(b − a) is finite, that is when b − a is not a non-positive integer and the real part of z goes to negative infinity, whereas the second term is not needed when Γ(a) is finite, that is, when a is a not a non-positive integer and the real part of z goes to positive infinity.

This gives (42) = 6 relations, given by identifying any two lines on the right hand side of In the notation above, M = M(a, b, z), M(a+) = M(a + 1, b, z), and so on.

There are similar relations for U. Kummer's functions are also related by Kummer's transformations: The following multiplication theorems hold true: In terms of Laguerre polynomials, Kummer's functions have several expansions, for example or Functions that can be expressed as special cases of the confluent hypergeometric function include: By applying a limiting argument to Gauss's continued fraction it can be shown that[5] and that this continued fraction converges uniformly to a meromorphic function of z in every bounded domain that does not include a pole.