Confluent hypergeometric function

File:Plot of the Kummer confluent hypergeometric function 1F1(a;b;z) with a=1 and b=2 and input z² with 1F1(1,2,z²) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1.svg

Plot of the Kummer confluent hypergeometric function 1F1(a;b;z) with a=1 and b=2 and input z² with 1F1(1,2,z²) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1

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 Template:Math, introduced by Template:Harvs, 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 Template:Math introduced by Template:Harvs, sometimes denoted by Template:Math, 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:

<math>z\frac{d^2w}{dz^2} + (b-z)\frac{dw}{dz} - aw = 0,</math>

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

Kummer's function of the first kind Template:Mvar is a generalized hypergeometric series introduced in Template:Harv, given by:

<math>M(a,b,z)=\sum_{n=0}^\infty \frac {a^{(n)} z^n} {b^{(n)} n!}={}_1F_1(a;b;z),</math>

where:

<math>a^{(0)}=1,</math>

<math>a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)\, ,</math>

is the rising factorial. Another common notation for this solution is Template:Math. Considered as a function of Template:Mvar, Template:Mvar, or Template:Mvar with the other two held constant, this defines an entire function of Template:Mvar or Template:Mvar, except when Template:Math As a function of Template:Mvar it is analytic except for poles at the non-positive integers.

Some values of Template:Mvar and Template:Mvar yield solutions that can be expressed in terms of other known functions. See #Special cases. When Template:Mvar 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

<math>M(a,c,z) = \lim_{b\to\infty}{}_2F_1(a,b;c;z/b)</math>

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 Template:Math. If we let Template:Math be

<math>w(z)=z^{1-b}v(z)</math>

then the differential equation gives

<math>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</math>

which, upon dividing out Template:Math and simplifying, becomes

<math>z\frac{d^2v}{dz^2}+(2-b-z)\frac{dv}{dz} - (a+1-b)v = 0.</math>

This means that Template:Math is a solution so long as Template:Mvar is not an integer greater than 1, just as Template:Math is a solution so long as Template:Mvar is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function Template:Math introduced by Template:Harvs, and sometimes denoted by Template:Math. It is a combination of the above two solutions, defined by

<math>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).</math>

Although this expression is undefined for integer Template:Mvar, it has the advantage that it can be extended to any integer Template:Mvar by continuity. Unlike Kummer's function which is an entire function of Template:Mvar, Template:Math usually has a singularity at zero. For example, if Template:Math and Template:Math then Template:Math is asymptotic to Template:Math as Template:Mvar goes to zero. But see #Special cases for some examples where it is an entire function (polynomial).

Note that the solution Template:Math to Kummer's equation is the same as the solution Template:Math, see #Kummer's transformation.

For most combinations of real or complex Template:Mvar and Template:Mvar, the functions Template:Math and Template:Math are independent, and if Template:Mvar is a non-positive integer, so Template:Math doesn't exist, then we may be able to use Template:Math as a second solution. But if Template:Mvar is a non-positive integer and Template:Mvar is not a non-positive integer, then Template:Math is a multiple of Template:Math. In that case as well, Template:Math can be used as a second solution if it exists and is different. But when Template:Mvar is an integer greater than 1, this solution doesn't exist, and if Template:Math then it exists but is a multiple of Template:Math and of Template:Math In those cases a second solution exists of the following form and is valid for any real or complex Template:Mvar and any positive integer Template:Mvar except when Template:Mvar is a positive integer less than Template:Mvar:

<math>M(a,b,z)\ln z+z^{1-b}\sum_{k=0}^\infty C_kz^k</math>

When a = 0 we can alternatively use:

<math>\int_{-\infty}^z(-u)^{-b}e^u\mathrm{d}u.</math>

When Template:Math this is the exponential integral Template:Math.

A similar problem occurs when Template:Math is a negative integer and Template:Mvar is an integer less than 1. In this case Template:Math doesn't exist, and Template:Math is a multiple of Template:Math A second solution is then of the form:

<math>z^{1-b}M(a+1-b,2-b,z)\ln z+\sum_{k=0}^\infty C_kz^k</math>

Other equations

Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as:

<math>z\frac{d^2w}{dz^2} +(b-z)\frac{dw}{dz} -\left(\sum_{m=0}^M a_m z^m\right)w = 0</math> <ref>Template:Cite journal</ref>

Note that for Template:Math 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 Template:Mvar, because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation:

<math>(A+Bz)\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0</math>

First we move the regular singular point to Template:Math by using the substitution of Template:Math, which converts the equation to:

<math>z\frac{d^2w}{dz^2} + (C+Dz)\frac{dw}{dz} +(E+Fz)w = 0</math>

with new values of Template:Mvar, and Template:Mvar. Next we use the substitution:

<math> z \mapsto \frac{1}{\sqrt{D^2-4F}} z</math>

and multiply the equation by the same factor, obtaining:

<math>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</math>

whose solution is

<math>\exp \left ( - \left (1+ \frac{D}{\sqrt{D^2-4F}} \right) \frac{z}{2} \right )w(z),</math>

where Template:Math is a solution to Kummer's equation with

<math>a=\left (1+ \frac{D}{\sqrt{D^2-4F}} \right)\frac{C}{2}-\frac{E}{\sqrt{D^2-4F}}, \qquad b = C.</math>

Note that the square root may give an imaginary or complex number. If it is zero, another solution must be used, namely

<math>\exp \left(-\tfrac{1}{2} Dz \right )w(z),</math>

where Template:Math is a confluent hypergeometric limit function satisfying

<math>zw(z)+Cw'(z)+\left(E-\tfrac{1}{2}CD \right)w(z)=0.</math>

As noted below, even the Bessel equation can be solved using confluent hypergeometric functions.

Integral representations

If Template:Math, Template:Math can be represented as an integral

<math>M(a,b,z)= \frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_0^1 e^{zu}u^{a-1}(1-u)^{b-a-1}\,du.</math>

thus Template:Math is the characteristic function of the beta distribution. For Template:Mvar with positive real part Template:Mvar can be obtained by the Laplace integral

<math>U(a,b,z) = \frac{1}{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt, \quad (\operatorname{Re}\ a>0) </math>

The integral defines a solution in the right half-plane Template:Math.

They can also be represented as Barnes integrals

<math>M(a,b,z) = \frac{1}{2\pi i}\frac{\Gamma(b)}{\Gamma(a)}\int_{-i\infty}^{i\infty} \frac{\Gamma(-s)\Gamma(a+s)}{\Gamma(b+s)}(-z)^sds</math>

where the contour passes to one side of the poles of Template:Math and to the other side of the poles of Template:Math.

Asymptotic behavior

If a solution to Kummer's equation is asymptotic to a power of Template:Mvar as Template:Math, then the power must be Template:Math. This is in fact the case for Tricomi's solution Template:Math. Its asymptotic behavior as Template:Math can be deduced from the integral representations. If Template:Math, 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 Template:Math:<ref>Template:Cite book.</ref>

<math>U(a,b,x)\sim x^{-a} \, _2F_0\left(a,a-b+1;\, ;-\frac 1 x\right),</math>

where <math>_2F_0(\cdot, \cdot; ;-1/x)</math> is a generalized hypergeometric series with 1 as leading term, which generally converges nowhere, but exists as a formal power series in Template:Math. This asymptotic expansion is also valid for complex Template:Mvar instead of real Template:Mvar, with Template:Math

The asymptotic behavior of Kummer's solution for large Template:Math is:

<math>M(a,b,z)\sim\Gamma(b)\left(\frac{e^zz^{a-b}}{\Gamma(a)}+\frac{(-z)^{-a}}{\Gamma(b-a)}\right)</math>

The powers of Template:Mvar are taken using Template:Math.<ref>This is derived from Abramowitz and Stegun (see reference below), page 508, where a full asymptotic series is given. They switch the sign of the exponent in Template:Math in the right half-plane but this is immaterial, as the term is negligible there or else Template:Mvar is an integer and the sign doesn't matter.</ref> The first term is not needed when Template:Math is finite, that is when Template:Math is not a non-positive integer and the real part of Template:Mvar goes to negative infinity, whereas the second term is not needed when Template:Math is finite, that is, when Template:Mvar is a not a non-positive integer and the real part of Template:Mvar goes to positive infinity.

There is always some solution to Kummer's equation asymptotic to Template:Math as Template:Math. Usually this will be a combination of both Template:Math and Template:Math but can also be expressed as Template:Math.

Relations

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

Contiguous relations

Given Template:Math, the four functions Template:Math are called contiguous to Template:Math. The function Template:Math can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of Template:Mvar, and Template:Mvar. This gives Template:Math relations, given by identifying any two lines on the right hand side of

<math>\begin{align}

z\frac{dM}{dz} = z\frac{a}{b}M(a+,b+) &=a(M(a+)-M)\\ &=(b-1)(M(b-)-M)\\ &=(b-a)M(a-)+(a-b+z)M\\ &=z(a-b)M(b+)/b +zM\\ \end{align}</math>

In the notation above, Template:Math, Template:Math, and so on.

Repeatedly applying these relations gives a linear relation between any three functions of the form Template:Math (and their higher derivatives), where Template:Mvar, Template:Mvar are integers.

There are similar relations for Template:Mvar.

Kummer's transformation

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

<math>M(a,b,z) = e^z\,M(b-a,b,-z)</math>

<math>U(a,b,z)=z^{1-b} U\left(1+a-b,2-b,z\right)</math>.

Multiplication theorem

The following multiplication theorems hold true:

<math>\begin{align}

U(a,b,z) &= e^{(1-t)z} \sum_{i=0} \frac{(t-1)^i z^i}{i!} U(a,b+i,z t)\\

        &= e^{(1-t)z} t^{b-1} \sum_{i=0} \frac{\left(1-\frac 1 t\right)^i}{i!} U(a-i,b-i,z t).

\end{align}</math>

Connection with Laguerre polynomials and similar representations

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

<math>M\left(a,b,\frac{x y}{x-1}\right) = (1-x)^a \cdot \sum_n\frac{a^{(n)}}{b^{(n)}}L_n^{(b-1)}(y)x^n</math> Template:Harv

or

<math>M\left( a,\, b,\, z \right) = \frac{\Gamma\left( 1 - a \right) \cdot \Gamma\left( b \right)}{\Gamma\left( b - a \right)} \cdot L_{-a}^{(b - 1)}\left( z \right)</math>[1]

Special cases

Functions that can be expressed as special cases of the confluent hypergeometric function include:

• Some elementary functions where the left-hand side is not defined when Template:Mvar is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation:

<math>M(0,b,z)=1</math>

<math>U(0,c,z)=1</math>

<math>M(b,b,z)=e^z</math>

<math>U(a,a,z)=e^z\int_z^\infty u^{-a}e^{-u}du</math> (a polynomial if Template:Mvar is a non-positive integer)

<math>\frac{U(1,b,z)}{\Gamma(b-1)}+\frac{M(1,b,z)}{\Gamma(b)}=z^{1-b}e^z</math>

<math>M(n,b,z)</math> for non-positive integer Template:Mvar is a generalized Laguerre polynomial.

<math>U(n,c,z)</math> for non-positive integer Template:Mvar is a multiple of a generalized Laguerre polynomial, equal to <math>\tfrac{\Gamma(1-c)}{\Gamma(n+1-c)}M(n,c,z)</math> when the latter exists.

<math>U(c-n,c,z)</math> when Template:Mvar is a positive integer is a closed form with powers of Template:Mvar, equal to <math>\tfrac{\Gamma(c-1)}{\Gamma(c-n)}z^{1-c}M(1-n,2-c,z)</math> when the latter exists.

<math>U(a,a+1,z)= z^{-a}</math>

<math>U(-n,-2n,z)</math> for non-negative integer Template:Mvar is a Bessel polynomial (see lower down).

<math>M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2</math> etc.

Using the contiguous relation <math>aM(a+)=(a+z)M+z(a-b)M(b+)/b</math> we get, for example, <math>M(2,1,z)=(1+z)e^z.</math>

• Bateman's function
• Bessel functions and many related functions such as Airy functions, Kelvin functions, Hankel functions. For example, in the special case Template:Math the function reduces to a Bessel function:

<math>{}_1F_1(a,2a,x)= e^{x/2}\, {}_0F_1 \left(; a+\tfrac{1}{2}; \tfrac{x^2}{16} \right) = e^{x/2} \left(\tfrac{x}{4}\right)^{1/2-a}\Gamma\left(a+\tfrac{1}{2}\right)I_{a-1/2}\left(\tfrac{x}{2}\right).</math>

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

<math>U(a,2a,x)= \frac{e^{x/2}}{\sqrt \pi} x^{1/2-a} K_{a-1/2} (x/2),</math>

When Template:Mvar is a non-positive integer, this equals Template:Math where Template:Mvar is a Bessel polynomial.

• The error function can be expressed as

<math>\mathrm{erf}(x)= \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt= \frac{2x}{\sqrt{\pi}}\ {}_1F_1\left(\tfrac{1}{2},\tfrac{3}{2},-x^2\right).</math>

• 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 Template:Math are solutions of Whittaker's equation that can be expressed in terms of Kummer functions Template:Mvar and Template:Mvar by

<math>M_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

<math>W_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

• The general Template:Mvar-th raw moment (Template:Mvar not necessarily an integer) can be expressed as<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

<math>\begin{align}

\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right] &= \frac{\left(2 \sigma^2\right)^{p/2} \Gamma\left(\tfrac{1+p}{2}\right)}{\sqrt \pi} \ {}_1F_1\left(-\tfrac p 2, \tfrac 1 2, -\tfrac{\mu^2}{2 \sigma^2}\right)\\ \operatorname{E} \left[N \left(\mu, \sigma^2 \right)^p \right] &= \left (-2 \sigma^2\right)^{p/2} U\left(-\tfrac p 2, \tfrac 1 2, -\tfrac{\mu^2}{2 \sigma^2} \right) \end{align}</math>

In the second formula the function's second branch cut can be chosen by multiplying with Template:Math.

Application to continued fractions

By applying a limiting argument to Gauss's continued fraction it can be shown that<ref>Template:Cite journal</ref>

<math>\frac{M(a+1,b+1,z)}{M(a,b,z)} = \cfrac{1}{1 - \cfrac{{\displaystyle\frac{b-a}{b(b+1)}z}}

{1 + \cfrac{{\displaystyle\frac{a+1}{(b+1)(b+2)}z}} {1 - \cfrac{{\displaystyle\frac{b-a+1}{(b+2)(b+3)}z}} {1 + \cfrac{{\displaystyle\frac{a+2}{(b+3)(b+4)}z}}{1 - \ddots}}}}} </math>

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

See also

• Composite Bézier curve

References

• Template:AS ref
• Template:Eom
• Template:Dlmf
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite journal
• Template:Cite book
• Template:Cite book

External links

• Confluent Hypergeometric Functions in NIST Digital Library of Mathematical Functions
• Kummer hypergeometric function on the Wolfram Functions site
• Tricomi hypergeometric function on the Wolfram Functions site