Confluent hypergeometric function


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. There are several common standard forms of confluent hypergeometric functions:
Kummer's equation may be written as:
with a regular singular point at and an irregular singular point at. It has two linearly independent solutions and.
Kummer's function M is a generalized hypergeometric series introduced in, given by:
where:
is the rising factorial. Another common notation for this solution is. Considered as a function of a, b, or with the other two held constant, this defines an entire function of a or z, except when As a function of it is analytic except for poles at the non-positive integers.
Some values of a and yield solutions that can be expressed in terms of other known functions. See #Special cases. When a is a non-positive integer then Kummer's function is a Laguerre polynomial.
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.
Since Kummer's equation is second order there must be another, independent, solution. The indicial equation of the method of Frobenius tells us that the lowest power of a power series solution to the Kummer equation is either 0 or. If we let w be
then the differential equation gives
which, upon dividing out and simplifying, becomes
This means that is a solution so long as b is not an integer greater than 1, just as is a solution so long as b is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function introduced by, and sometimes denoted by. It is a combination of the above two solutions, defined by
Although this expression is undefined for integer, it has the advantage that it can be extended to any integer by continuity. Unlike Kummer's function which is an entire function of z, U usually has a singularity at zero. For example, if b=0 and a≠0 then is asymptotic to as z goes to zero. But see #Special cases for some examples where it is an entire function.
Note that the solution to Kummer's equation is the same as the solution
For most combinations of real a and b, the functions and are independent, and if b is a non-positive integer then we may be able to use as a second solution. But if is a non-positive integer and is not a non-positive integer, then is a multiple of. In that case as well, can be used as a second solution if it exists and is different. But when b is an integer greater than 1 this solution doesn't exist, and if then it exists but is a multiple of and of In those cases a second solution exists of the form :
When a = 0 we can alternatively use:
When this is the exponential integral E1.
A similar problem occurs when a−b is a negative integer and b is an integer less than 1. In this case doesn't exist, and is a multiple of A second solution is then of the form:

Other equations

Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as:
Thus Confluent Hypergeometric Functions can be used to solve "most" second-order ordinary differential equations whose variable coefficients are all linear functions of ; because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation:
First we move the regular singular point to by using the substitution of which converts the equation to:
with new values of, and. Next we use the substitution:
and multiply the equation by the same factor, obtaining:
whose solution is
where is a solution to Kummer's equation with
Note that the square root may give an imaginary number. If it is zero, another solution must be used, namely
where is a confluent hypergeometric limit function satisfying
As noted lower down, even the Bessel equation can be solved using confluent hypergeometric functions.

Integral representations

If Re b > Re a > 0, can be represented as an integral
thus is the characteristic function of the beta distribution. For a with positive real part can be obtained by the Laplace integral
The integral defines a solution in the right half-plane .
They can also be represented as Barnes integrals
where the contour passes to one side of the poles of and to the other side of the poles of.

Asymptotic behavior

If a solution to Kummer's equation is asymptotic to a power of as, then the power must be. This is in fact the case for Tricomi's solution. Its asymptotic behavior as can be deduced from the integral representations. If, 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 :
where is a generalized hypergeometric series, which generally converges nowhere but exists as a formal power series in 1/x. This asymptotic expansion is also valid for complex instead of real x, with
The asymptotic behavior of Kummer's solution for large |z| is:
The powers of are taken using. The first term is not needed when is finite and the real part of goes to negative infinity, whereas the second term is not needed when is finite and the real part of goes to positive infinity.
There is always some solution to Kummer's equation asymptotic to as. Usually this will be a combination of both and but can also be expressed as.

Relations

There are many relations between Kummer functions for various arguments and their derivatives. This section gives a few typical examples.

Contiguous relations

Given, the four functions are called contiguous to. The function can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of, and. This gives =6 relations, given by identifying any two lines on the right hand side of
In the notation above, M = M, M = M, and so on.
Repeatedly applying these relations gives a linear relation between any three functions of the form , where m, n are integers.
There are similar relations for U.

Kummer's transformation

Kummer's functions are also related by Kummer's transformations:

Multiplication theorem

The following multiplication theorems hold true:

Connection with Laguerre polynomials and similar representations

In terms of Laguerre polynomials, Kummer's functions have several expansions, for example

Special cases

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
and that this continued fraction converges uniformly to a meromorphic function of in every bounded domain that does not include a pole.