Confluent hypergeometric function

Kummer's equation may be written as:

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

with a regular singular point at z = 0 and an irregular singular point at z = ∞. It has two (usually) linearly independent solutions M(a, b, z) and U(a, b, z).

Kummer's function of the first kind M is a generalized hypergeometric series introduced in (Kummer 1837), given by:

${\displaystyle M(a,b,z)=\sum _{n=0}^{\infty }{\frac {a^{(n)}z^{n}}{b^{(n)}n!}}={}_{1}F_{1}(a;b;z),}$

where:

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

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

is the rising factorial. Another common notation for this solution is Φ(a, b, z). Considered as a function of a, b, or z with the other two held constant, this defines an entire function of a or z, except when b = 0, −1, −2, ... As a function of b it is analytic except for poles at the non-positive integers.

Some values of a and b 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

${\displaystyle M(a,c,z)=\lim _{b\to \infty }{}_{2}F_{1}(a,b;c;z/b)}$

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. 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 1 − b. If we let w(z) be

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

then the differential equation gives

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

which, upon dividing out z^1−b and simplifying, becomes

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

This means that z^1−bM(a + 1 − b, 2 − b, z) is a solution so long as b is not an integer greater than 1, just as M(a, b, z) is a solution so long as b is not an integer less than 1. We can also use the Tricomi confluent hypergeometric function U(a, b, z) introduced by Francesco Tricomi (1947), and sometimes denoted by Ψ(a; b; z). It is a combination of the above two solutions, defined by

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

Although this expression is undefined for integer b, it has the advantage that it can be extended to any integer b 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 Γ(a+1)U(a, b, z) − 1 is asymptotic to az ln z as z goes to zero. But see #Special cases for some examples where it is an entire function (polynomial).

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

For most combinations of real or complex a and b, the functions M(a, b, z) and U(a, b, z) are independent, and if b is a non-positive integer, so M(a, b, z) doesn't exist, then we may be able to use z^1−bM(a+1−b, 2−b, z) as a second solution. But if a is a non-positive integer and b is not a non-positive integer, then U(z) is a multiple of M(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 when b is an integer greater than 1, this solution doesn't exist, and if b = 1 then it exists but is a multiple of U(a, b, z) and of M(a, b, z) In those cases a second solution exists of the following form and is valid for any real or complex a and any positive integer b except when a is a positive integer less than b:

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

When a = 0 we can alternatively use:

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

When b = 1 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 M(a, b, z) doesn't exist, and U(a, b, z) is a multiple of z^1−bM(a+1−b, 2−b, z). A second solution is then of the form:

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

If Re b > Re a > 0, M(a, b, z) can be represented as an integral

${\displaystyle M(a,b,z)={\frac {\Gamma (b)}{\Gamma (a)\Gamma (b-a)}}\int _{0}^{1}e^{zu}u^{a-1}(1-u)^{b-a-1}\,du.}$

thus M(a, a+b, it) is the characteristic function of the beta distribution. For a with positive real part U can be obtained by the Laplace integral

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

The integral defines a solution in the right half-plane 0 < Re z < π/2.

They can also be represented as Barnes integrals

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

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

If a solution to Kummer's equation is asymptotic to a power of z as z → ∞, then the power must be −a. This is in fact the case for Tricomi's solution U(a, b, z). Its asymptotic behavior as z → ∞ can be deduced from the integral representations. If z = x ∈ R, 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 x → ∞:[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 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 z instead of real x, with |arg z| < 3π/2.

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

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

The powers of z are taken using −3π/2 < arg z ≤ π/2.[3] The first term is not needed when Γ(b − a) is finite, that is when b − a is not a non-positive integer and the real part of z goes to negative infinity, whereas the second term is not needed when Γ(a) is finite, that is, when a is a not a non-positive integer and the real part of z goes to positive infinity.

There is always some solution to Kummer's equation asymptotic to e^zz^a−b as z → −∞. Usually this will be a combination of both M(a, b, z) and U(a, b, z) but can also be expressed as e^z (−1)^a-b U(b − a, b, −z).

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

The following multiplication theorems hold true:

${\displaystyle {\begin{aligned}U(a,b,z)&=e^{(1-t)z}\sum _{i=0}{\frac {(t-1)^{i}z^{i}}{i!}}U(a,b+i,zt)\\&=e^{(1-t)z}t^{b-1}\sum _{i=0}{\frac {\left(1-{\frac {1}{t}}\right)^{i}}{i!}}U(a-i,b-i,zt).\end{aligned}}}$

In terms of Laguerre 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)

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 b is a non-positive integer, but the right-hand side is still a solution of the corresponding Kummer equation:

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

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

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

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

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

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

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

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

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

${\displaystyle U(-n,-2n,z)}$ for non-negative integer n 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 as Airy functions, Kelvin functions, Hankel functions. For example, in the special case b = 2a the function reduces to a Bessel function:

${\displaystyle {}_{1}F_{1}(a,2a,x)=e^{x/2}\,{}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)=e^{x/2}\left({\tfrac {x}{4}}\right)^{1/2-a}\Gamma \left(a+{\tfrac {1}{2}}\right)I_{a-1/2}\left({\tfrac {x}{2}}\right).}$

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

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

When a is a non-positive integer, this equals 2^−aθ_−a(x/2) where θ is a Bessel polynomial.

• The error function can be expressed as

${\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt={\frac {2x}{\sqrt {\pi }}}\ {}_{1}F_{1}\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).}$

• 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 M_κ,μ(z), W_κ,μ(z) are solutions of Whittaker's equation that can be expressed in terms of Kummer functions M and U by

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

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

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

${\displaystyle {\begin{aligned}\operatorname {E} \left[\left|N\left(\mu ,\sigma ^{2}\right)\right|^{p}\right]&={\frac {\left(2\sigma ^{2}\right)^{p/2}\Gamma \left({\tfrac {1+p}{2}}\right)}{\sqrt {\pi }}}\ {}_{1}F_{1}\left(-{\tfrac {p}{2}},{\tfrac {1}{2}},-{\tfrac {\mu ^{2}}{2\sigma ^{2}}}\right)\\\operatorname {E} \left[N\left(\mu ,\sigma ^{2}\right)^{p}\right]&=\left(-2\sigma ^{2}\right)^{p/2}U\left(-{\tfrac {p}{2}},{\tfrac {1}{2}},-{\tfrac {\mu ^{2}}{2\sigma ^{2}}}\right)\end{aligned}}}$

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

By applying a limiting argument to Gauss's continued fraction it can be shown that

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

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

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• Chistova, E.A. (2001) [1994], "Confluent hypergeometric function", Encyclopedia of Mathematics, EMS Press
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• 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). 1. Rome: Edizioni cremonese. ISBN 978-88-7083-449-9. MR 0076936.
• Oldham, K.B.; Myland, J.; Spanier, J. (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. An Atlas of Functions. Springer New York. ISBN 978-0-387-48807-3. Retrieved 2017-08-23.

• 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