Inmathematics, aconfluenthypergeometric function is a solution of aconfluent hypergeometric equation, which is a degenerate form of ahypergeometric differential equation where two of the threeregular singularities merge into anirregular singularity. The termconfluent 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) functionM(a,b,z), introduced byKummer (1837), is a solution toKummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelatedKummer's function bearing the same name.
• Tricomi's (confluent hypergeometric) functionU(a,b,z) introduced byFrancesco Tricomi (1947), sometimes denoted byΨ(a;b;z), is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind.
• Whittaker functions (forEdmund Taylor Whittaker) are solutions toWhittaker's equation.
• Coulomb wave functions are solutions to theCoulomb 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 may be written as:

${\displaystyle z\frac{d^{2}w}{d{z}^2}+(b-z)\frac{dw}{dz}-aw=0,}$

with a regular singular point atz = 0 and an irregular singular point atz = ∞. It has two (usually)linearly independent solutionsM(a,b,z) andU(a,b,z).

Kummer's function of the first kindM is ageneralized hypergeometric series introduced in (Kummer 1837), given by:

${\displaystyle M(a,b,z)=\sum _{n=0}^{\mathrm{\infty }}\frac{a^{(n)}{z}^n}{b^{(n)}n!}={}_1{F}_1(a;b;z),}$

where:

${\displaystyle a^{(0)}=1,}$

${\displaystyle a^{(n)}=a(a+1)(a+2)\cdots (a+n-1),}$

is therising factorial. Another common notation for this solution isΦ(a,b,z). Considered as a function ofa,b, orz with the other two held constant, this defines anentire function ofa orz, except whenb = 0, −1, −2, ... As a function ofb it isanalytic except for poles at the non-positive integers.

Some values ofa andb yield solutions that can be expressed in terms of other known functions. See#Special cases. Whena is a non-positive integer, then Kummer's function (if it is defined) is a generalizedLaguerre polynomial.

Just as the confluent differential equation is a limit of thehypergeometric 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 thehypergeometric function

${\displaystyle M(a,c,z)=\underset{b\to \mathrm{\infty }}{lim}{}_2{F}_1(a,b;c;z/b)}$

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. Theindicial equation of the method of Frobenius tells us that the lowest power of apower series solution to the Kummer equation is either 0 or1 −b. If we letw(z) be

${\displaystyle w(z)=z^{1-b}v(z)}$

then the differential equation gives

${\displaystyle z^{2-b}\frac{d^{2}v}{d{z}^2}+2(1-b)z^{1-b}\frac{dv}{dz}-b(1-b)z^{-b}v+(b-z)\left[z^{1-b}\frac{dv}{dz}+(1-b)z^{-b}v\right]-a{z}^{1-b}v=0}$

which, upon dividing outz^1−b and simplifying, becomes

${\displaystyle z\frac{d^{2}v}{d{z}^2}+(2-b-z)\frac{dv}{dz}-(a+1-b)v=0.}$

This means thatz^1−bM(a + 1 −b, 2 −b,z) is a solution so long asb is not an integer greater than 1, just asM(a,b,z) is a solution so long asb is not an integer less than 1. We can also use the Tricomi confluent hypergeometric functionU(a,b,z) introduced byFrancesco Tricomi (1947), and sometimes denoted byΨ(a;b;z). It is a combination of the above two solutions, defined by

${\displaystyle U(a,b,z)=\frac{\mathrm{\Gamma }(1-b)}{\mathrm{\Gamma }(a+1-b)}M(a,b,z)+\frac{\mathrm{\Gamma }(b-1)}{\mathrm{\Gamma }(a)}{z}^{1-b}M(a+1-b,2-b,z).}$

Although this expression is undefined for integerb, it has the advantage that it can be extended to any integerb by continuity. Unlike Kummer's function which is anentire function ofz,U(z) usually has asingularity at zero. For example, ifb = 0 anda ≠ 0 thenΓ(a+1)U(a,b,z) − 1 is asymptotic toaz lnz asz goes to zero. But see#Special cases for some examples where it is an entire function (polynomial).

Note that the solutionz^1−bU(a + 1 −b, 2 −b,z) to Kummer's equation is the same as the solutionU(a,b,z), see#Kummer's transformation.

For most combinations of real or complexa andb, the functionsM(a,b,z) andU(a,b,z) are independent, and ifb is a non-positive integer, soM(a,b,z) doesn't exist, then we may be able to usez^1−bM(a+1−b, 2−b,z) as a second solution. But ifa is a non-positive integer andb is not a non-positive integer, thenU(z) is a multiple ofM(z). In that case as well,z^1−bM(a+1−b, 2−b,z) can be used as a second solution if it exists and is different. But whenb is an integer greater than 1, this solution doesn't exist, and ifb = 1 then it exists but is a multiple ofU(a,b,z) and ofM(a,b,z) In those cases a second solution exists of the following form and is valid for any real or complexa and any positive integerb except whena is a positive integer less thanb:

${\displaystyle M(a,b,z)\ln z+z^{1-b}\sum _{k=0}^{\mathrm{\infty }}{C}_k{z}^k}$

Whena = 0 we can alternatively use:

${\displaystyle {\int }_{-\mathrm{\infty }}^z(-u{)}^{-b}{e}^u\mathrm{d}u.}$

Whenb = 1 this is theexponential integralE₁(−z).

A similar problem occurs whena−b is a negative integer andb is an integer less than 1. In this caseM(a,b,z) doesn't exist, andU(a,b,z) is a multiple ofz^1−bM(a+1−b, 2−b,z). A second solution is then of the form:

${\displaystyle z^{1-b}M(a+1-b,2-b,z)\ln z+\sum _{k=0}^{\mathrm{\infty }}{C}_k{z}^k}$

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

${\displaystyle z\frac{d^{2}w}{d{z}^2}+(b-z)\frac{dw}{dz}-\left(\sum _{m=0}^M{a}_m{z}^m\right)w=0}$^[1]

Note that forM = 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 ofz, because they can be transformed to the Extended Confluent Hypergeometric Equation. Consider the equation:

${\displaystyle (A+Bz)\frac{d^{2}w}{d{z}^2}+(C+Dz)\frac{dw}{dz}+(E+Fz)w=0}$

First we move theregular singular point to0 by using the substitution ofA +Bz ↦z, which converts the equation to:

${\displaystyle z\frac{d^{2}w}{d{z}^2}+(C+Dz)\frac{dw}{dz}+(E+Fz)w=0}$

with new values ofC, D, E, andF. Next we use the substitution:

${\displaystyle z\mapsto \frac{1}{\sqrt{D^2-4F}}z}$

and multiply the equation by the same factor, obtaining:

${\displaystyle z\frac{d^{2}w}{d{z}^2}+\left(C+\frac{D}{\sqrt{D^2-4F}}z\right)\frac{dw}{dz}+\left(\frac{E}{\sqrt{D^2-4F}}+\frac{F}{D^2-4F}z\right)w=0}$

whose solution is

${\displaystyle \exp \left(-\left(1+\frac{D}{\sqrt{D^2-4F}}\right)\frac{z}{2}\right)w(z),}$

wherew(z) is a solution to Kummer's equation with

${\displaystyle a=\left(1+\frac{D}{\sqrt{D^2-4F}}\right)\frac{C}{2}-\frac{E}{\sqrt{D^2-4F}},b=C.}$

Note that thesquare root may give an imaginary orcomplex number. If it is zero, another solution must be used, namely

${\displaystyle \exp \left(-\frac{1}{2}Dz\right)w(z),}$

wherew(z) is aconfluent hypergeometric limit function satisfying

${\displaystyle z{w}^{″}(z)+C{w}^{\prime }(z)+\left(E-\frac{1}{2}CD\right)w(z)=0.}$

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

IfReb > Rea > 0,M(a,b,z) can be represented as an integral

${\displaystyle M(a,b,z)=\frac{\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b-a)}{\int }_0^1{e}^{zu}{u}^{a-1}(1-u{)}^{b-a-1}du.}$

thusM(a,a+b,it) is thecharacteristic function of thebeta distribution. Fora with positive real partU can be obtained by theLaplace integral

${\displaystyle U(a,b,z)=\frac{1}{\mathrm{\Gamma }(a)}{\int }_0^{\mathrm{\infty }}{e}^{-zt}{t}^{a-1}(1+t{)}^{b-a-1}dt,(\mathrm{Re}a>0)}$

The integral defines a solution in the right half-planeRez > 0.

They can also be represented asBarnes integrals

${\displaystyle M(a,b,z)=\frac{1}{2\pi i}\frac{\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(a)}{\int }_{-i\mathrm{\infty }}^{i\mathrm{\infty }}\frac{\mathrm{\Gamma }(-s)\mathrm{\Gamma }(a+s)}{\mathrm{\Gamma }(b+s)}(-z{)}^{s}ds}$

where the contour passes to one side of the poles ofΓ(−s) and to the other side of the poles ofΓ(a +s).

If a solution to Kummer's equation is asymptotic to a power ofz asz → ∞, then the power must be−a. This is in fact the case for Tricomi's solutionU(a,b,z). Itsasymptotic behavior asz → ∞ can be deduced from the integral representations. Ifz =x ∈R, then making a change of variables in the integral followed by expanding thebinomial series and integrating it formally term by term gives rise to anasymptotic series expansion, valid asx → ∞:^[2]

${\displaystyle U(a,b,x)\sim x^{-a}{}_2{F}_0\left(a,a-b+1;;-\frac{1}{x}\right),}$

where${\displaystyle {}_2{F}_0(\cdot ,\cdot ;;-1/x)}$ is ageneralized hypergeometric series with 1 as leading term, which generally converges nowhere, but exists as aformal power series in1/x. Thisasymptotic expansion is also valid for complexz instead of realx, with|argz| < 3π/2.

The asymptotic behavior of Kummer's solution for large|z| is:

${\displaystyle M(a,b,z)\sim \mathrm{\Gamma }(b)\left(\frac{e^z{z}^{a-b}}{\mathrm{\Gamma }(a)}+\frac{(-z{)}^{-a}}{\mathrm{\Gamma }(b-a)}\right)}$

The powers ofz are taken using−3π/2 < argz ≤π/2.^[3] The first term is not needed whenΓ(b −a) is finite, that is whenb −a is not a non-positive integer and the real part ofz goes to negative infinity, whereas the second term is not needed whenΓ(a) is finite, that is, whena is a not a non-positive integer and the real part ofz goes to positive infinity.

There is always some solution to Kummer's equation asymptotic toe^zz^a−b asz → −∞. Usually this will be a combination of bothM(a,b,z) andU(a,b,z) but can also be expressed ase^z (−1)^a-bU(b −a,b, −z).

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

GivenM(a,b,z), the four functionsM(a ± 1,b,z),M(a,b ± 1,z) are called contiguous toM(a,b,z). The functionM(a,b,z) can be written as alinear combination of any two of its contiguous functions, with rational coefficients in terms ofa, b, andz. This gives(⁴
₂) = 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 formM(a +m,b +n,z) (and their higher derivatives), wherem,n are integers.

There are similar relations forU.

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

${\displaystyle M(a,b,z)=e^{z}M(b-a,b,-z)}$

${\displaystyle U(a,b,z)=z^{1-b}U\left(1+a-b,2-b,z\right)}$.

The followingmultiplication theorems hold true:

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

${\displaystyle M\left(a,b,\frac{xy}{x-1}\right)=(1-x{)}^a\cdot \sum _n\frac{a^{(n)}}{b^{(n)}}{L}_n^{(b-1)}(y)x^n}$ (Erdélyi et al. 1953, 6.12)

or

${\displaystyle M\left(a,b,z\right)=\frac{\mathrm{\Gamma }\left(1-a\right)\cdot \mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(b-a\right)}\cdot L_{-a}^{(b-1)}\left(z\right)}$[1]

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

• Someelementary functions where the left-hand side is not defined whenb is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation:

${\displaystyle M(0,b,z)=1}$

${\displaystyle U(0,c,z)=1}$

${\displaystyle M(b,b,z)=e^z}$

${\displaystyle U(a,a,z)=e^z{\int }_z^{\mathrm{\infty }}{u}^{-a}{e}^{-u}du}$ (a polynomial ifa is a non-positive integer)

${\displaystyle \frac{U(1,b,z)}{\mathrm{\Gamma }(b-1)}+\frac{M(1,b,z)}{\mathrm{\Gamma }(b)}=z^{1-b}{e}^z}$

${\displaystyle M(n,b,z)}$ for non-positive integern is ageneralized Laguerre polynomial.

${\displaystyle U(n,c,z)}$ for non-positive integern is a multiple of a generalized Laguerre polynomial, equal to${\displaystyle \frac{\mathrm{\Gamma }(1-c)}{\mathrm{\Gamma }(n+1-c)}M(n,c,z)}$ when the latter exists.

${\displaystyle U(c-n,c,z)}$ whenn is a positive integer is a closed form with powers ofz, equal to${\displaystyle \frac{\mathrm{\Gamma }(c-1)}{\mathrm{\Gamma }(c-n)}{z}^{1-c}M(1-n,2-c,z)}$ when the latter exists.

${\displaystyle U(a,a+1,z)=z^{-a}}$

${\displaystyle U(-n,-2n,z)}$ for non-negative integern is a Bessel polynomial (see lower down).

${\displaystyle M(1,2,z)=(e^z-1)/z,M(1,3,z)=2!(e^z-1-z)/z^2}$ etc.

Using the contiguous relation${\displaystyle aM(a+)=(a+z)M+z(a-b)M(b+)/b}$ we get, for example,${\displaystyle M(2,1,z)=(1+z)e^z.}$

• Bateman's function
• Bessel functions and many related functions such asAiry functions,Kelvin functions,Hankel functions. For example, in the special caseb = 2a the function reduces to aBessel function:

${\displaystyle {}_1{F}_1(a,2a,x)=e^{x/2}{}_0{F}_1\left(;a+\frac{1}{2};\frac{x^2}{16}\right)=e^{x/2}{\left(\frac{x}{4}\right)}^{1/2-a}\mathrm{\Gamma }\left(a+\frac{1}{2}\right)I_{a-1/2}\left(\frac{x}{2}\right).}$

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

${\displaystyle U(a,2a,x)=\frac{e^{x/2}}{\sqrt{\pi }}{x}^{1/2-a}{K}_{a-1/2}(x/2),}$

Whena is a non-positive integer, this equals2^−aθ_−a(x/2) whereθ is aBessel polynomial.

• Theerror function can be expressed as

${\displaystyle \mathrm{e}\mathrm{r}\mathrm{f}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt=\frac{2x}{\sqrt{\pi }}{}_1{F}_1\left(\frac{1}{2},\frac{3}{2},-x^2\right).}$

• Coulomb wave function
• Cunningham functions
• Exponential integral and related functions such as thesine integral,logarithmic integral
• Hermite polynomials
• Incomplete gamma function
• Laguerre polynomials
• Parabolic cylinder function (or Weber function)
• Poisson–Charlier function
• Toronto functions
• Whittaker functionsM_κ,μ(z),W_κ,μ(z) are solutions ofWhittaker's equation that can be expressed in terms of Kummer functionsM andU by

${\displaystyle M_{\kappa ,\mu }(z)=e^{-\frac{z}{2}}{z}^{\mu +\frac{1}{2}}M\left(\mu -\kappa +\frac{1}{2},1+2\mu ;z\right)}$

${\displaystyle W_{\kappa ,\mu }(z)=e^{-\frac{z}{2}}{z}^{\mu +\frac{1}{2}}U\left(\mu -\kappa +\frac{1}{2},1+2\mu ;z\right)}$

• The generalp-th raw moment (p not necessarily an integer) can be expressed as^[4]

In the second formula the function's secondbranch cut can be chosen by multiplying with(−1)^p.

By applying a limiting argument toGauss's continued fraction it can be shown that^[5]

${\displaystyle \frac{M(a+1,b+1,z)}{M(a,b,z)}=\frac{1}{1-\frac{{\displaystyle \frac{b-a}{b(b+1)}z}}{1+\frac{{\displaystyle \frac{a+1}{(b+1)(b+2)}z}}{1-\frac{{\displaystyle \frac{b-a+1}{(b+2)(b+3)}z}}{1+\frac{{\displaystyle \frac{a+2}{(b+3)(b+4)}z}}{1-\ddots }}}}}}$

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

• Composite Bézier curve

• Abramowitz, Milton;Stegun, Irene Ann, eds. (1983) [June 1964]."Chapter 13".Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 504.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253.
• Chistova, E.A. (2001) [1994],"Confluent hypergeometric function",Encyclopedia of Mathematics,EMS Press
• Daalhuis, Adri B. Olde (2010),"Confluent hypergeometric function", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.
• Erdélyi, Arthur;Magnus, Wilhelm; Oberhettinger, Fritz & Tricomi, Francesco G. (1953).Higher transcendental functions. Vol. I. New York–Toronto–London: McGraw–Hill Book Company, Inc.MR 0058756.
• Kummer, Ernst Eduard (1837)."De integralibus quibusdam definitis et seriebus infinitis".Journal für die reine und angewandte Mathematik (in Latin).1837 (17):228–242.doi:10.1515/crll.1837.17.228.ISSN 0075-4102.S2CID 121351583.
• Slater, Lucy Joan (1960).Confluent hypergeometric functions. Cambridge, UK: Cambridge University Press.MR 0107026.
• Tricomi, Francesco G. (1947)."Sulle funzioni ipergeometriche confluenti".Annali di Matematica Pura ed Applicata. Series 4 (in Italian).26:141–175.doi:10.1007/bf02415375.ISSN 0003-4622.MR 0029451.S2CID 119860549.
• Tricomi, Francesco G. (1954).Funzioni ipergeometriche confluenti. Consiglio Nazionale Delle Ricerche Monografie Matematiche (in Italian). Vol. 1. Rome: Edizioni cremonese.ISBN 978-88-7083-449-9.MR 0076936.{{cite book}}:ISBN / Date incompatibility (help)
• Oldham, K.B.; Myland, J.; Spanier, J. (2010).An Atlas of Functions: with Equator, the Atlas Function Calculator. Springer New York.ISBN 978-0-387-48807-3. Retrieved2017-08-23.

• 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

• Hypergeometric functions
• Special hypergeometric functions
• Special functions

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