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. (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 (confluent hypergeometric) function , introduced by , is a solution to Kummer's  differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name.
 Tricomi's (confluent hypergeometric) function  introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind.
 Whittaker functions (for Edmund Taylor Whittaker) are  solutions to Whittaker's equation.
 Coulomb wave functions are solutions to the Coulomb wave equation. The Kummer functions,  Whittaker functions, and Coulomb wave functions are essentially the same, and differ from each other only by elementary functions and change of variables.

Kummer's equation
Kummer's equation may be written as:

with a regular singular point at  and an irregular singular point at . It has two (usually) linearly independent solutions  and .

Kummer's function (of the first kind) 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 (if it is defined) is a (generalized) 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(z) 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(z) 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 (polynomial).

Note that the solution  to Kummer's equation is the same as the solution  (See #Kummer's transformation below.)

For most combinations of real (or complex) a and b, the functions  and  are independent, and if b is a non-positive integer (so  doesn't exist) 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 (valid for any real or complex a and any positive integer b except when a is a positive integer less than b):

When a = 0 we can alternatively use:

When  this is the exponential integral E1(-z).

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:
 

{NB that for m=0 (or when the summation involves just one term), it reduces to the conventional Confluent Hypergeometric Equation}

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 (or complex) 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 (with 1 as leading term), 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 (that is, when  is not a non-positive integer) and the real part of  goes to negative infinity, whereas the second term is not needed when  is finite (that is, when a is a not a non-positive integer) 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(a, b, z), M(a+) = M(a + 1, b, z), and so on.

Repeatedly applying these relations gives a linear relation between any three functions of the form  (and their higher derivatives), 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:
Some elementary functions (the left-hand side is not defined when  is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation):

 (a polynomial if a is a non-positive integer)

 for non-positive integer n is a generalized Laguerre polynomial.
 for non-positive integer n is a multiple of a generalized Laguerre polynomial, equal to  when the latter exists.
 when n is a positive integer is a closed form with powers of z, equal to  when the latter exists.

 for non-negative integer n is a Bessel polynomial (see lower down).
 etc.
Using the contiguous relation  we get, for example, 
Bateman's function
Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. For example, in the special case  the function reduces to a Bessel function:

This identity is sometimes also referred to as Kummer's second transformation. Similarly

When  is a non-positive integer, this equals  where  is a Bessel polynomial.
 The error function can be expressed as

Coulomb wave function
Cunningham functions
Exponential integral and related functions such as the sine integral, logarithmic integral
Hermite polynomials
Incomplete gamma function
Laguerre polynomials
Parabolic cylinder function (or Weber function)
Poisson–Charlier function
Toronto functions
Whittaker functions  are solutions of Whittaker's equation that can be expressed in terms of Kummer functions M and U by

 The general -th raw moment ( not necessarily an integer) can be expressed as
  
In the second formula the function's second branch cut can be chosen by multiplying with .

Application to continued fractions

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.

Notes