Kummer's equation may be written as: :z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0, 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 is a
generalized hypergeometric series introduced in , given by: :M(a,b,z)=\sum_{n=0}^\infty \frac {a^{(n)} z^n} {b^{(n)} n!}={}_1F_1(a;b;z), where: : a^{(0)}=1, : a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)\, , is the
rising factorial. Another common notation for this solution is . Considered as a function of , , or with the other two held constant, this defines an
entire function of or , except when As a function of it is
analytic except for poles at the non-positive integers. Some values of and yield solutions that can be expressed in terms of other known functions. See #Special cases. When 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 :M(a,c,z) = \lim_{b\to\infty}{}_2F_1(a,b;c;z/b) 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 be :w(z)=z^{1-b}v(z) then the differential equation gives :z^{2-b}\frac{d^2v}{dz^2}+2(1-b)z^{1-b}\frac{dv}{dz}-b(1-b)z^{-b}v + (b-z)\left[z^{1-b}\frac{dv}{dz}+(1-b)z^{-b}v\right] - az^{1-b}v = 0 which, upon dividing out and simplifying, becomes z\frac{d^2v}{dz^2}+2(1-b)\frac{dv}{dz}-b(1-b)z^{-1}v + (b-z)\left[\frac{dv}{dz}+(1-b)z^{-1}v\right] - av = 0--> :z\frac{d^2v}{dz^2}+(2-b-z)\frac{dv}{dz} - (a+1-b)v = 0. This means that is a solution so long as is not an integer greater than 1, just as is a solution so long as 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 :U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a+1-b)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a+1-b,2-b,z). 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 , usually has a
singularity at zero. For example, if and then is asymptotic to as 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. For most combinations of real or complex and , the functions and are independent, and if 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 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 following form and is valid for any real or complex and any positive integer except when is a positive integer less than : :M(a,b,z)\ln z+z^{1-b}\sum_{k=0}^\infty C_kz^k When
a = 0 we can alternatively use: :\int_{-\infty}^z(-u)^{-b}e^u\mathrm{d}u. When this is the
exponential integral . A similar problem occurs when is a negative integer and 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: :z^{1-b}M(a+1-b,2-b,z)\ln z+\sum_{k=0}^\infty C_kz^k
Other equations Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as: :z\frac{d^2w}{dz^2} +(b-z)\frac{dw}{dz} -\left(\sum_{m=0}^M a_m z^m\right)w = 0 Note that for 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: :(A+Bz)\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0 First we move the
regular singular point to by using the substitution of , which converts the equation to: :z\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0 with new values of , and . Next we use the substitution: : z \mapsto \frac{1}{\sqrt{D^2-4F}} z and multiply the equation by the same factor, obtaining: :z\frac{d^2w}{dz^2}+\left(C+\frac{D}{\sqrt{D^2-4F}}z\right)\frac{dw}{dz}+\left(\frac{E}{\sqrt{D^2-4F}}+\frac{F}{D^2-4F}z\right)w=0 whose solution is :\exp \left ( - \left (1+ \frac{D}{\sqrt{D^2-4F}} \right) \frac{z}{2} \right )w(z), where is a solution to Kummer's equation with :a=\left (1+ \frac{D}{\sqrt{D^2-4F}} \right)\frac{C}{2}-\frac{E}{\sqrt{D^2-4F}}, \qquad b = C. Note that the
square root may give an imaginary or
complex number. If it is zero, another solution must be used, namely :\exp \left(-\tfrac{1}{2} Dz \right )w(z), where is a
confluent hypergeometric limit function satisfying :zw''(z)+Cw'(z)+\left(E-\tfrac{1}{2}CD \right)w(z)=0. As noted below, even the
Bessel equation can be solved using confluent hypergeometric functions. ==Integral representations==