Computing the confluent hypergeometric function, M(a,b,x)

Summary. The confluent hypergeometric function, M(a,b,x), arises naturally in both statistics and physics. Although analytically well-behaved, extreme but practically useful combinations of parameters create extreme computational difficulties. A brief review of known analytic and computational results highlights some difficult regions, including , with x much larger than b. Existing power series and integral representations may fail to converge numerically, while asymptotic series representations may diverge before achieving the accuracy desired. Continued fraction representations help somewhat. Variable precision can circumvent the problem, but with reductions in speed and convenience. In some cases, known analytic properties allow transforming a difficult computation into an easier one. The combination of existing computational forms and transformations still leaves gaps. For , two new power series, in terms of Gamma and Beta cumulative distribution functions respectively, help in some cases. Numerical evaluations highlight the abilities and limitations of existing and new methods. Overall, a rational approximation due to Luke and the new Gamma-based series provide the best performance. Received August 16, 1999 / Revised version received September 15, 2000 / Published online May 4, 2001     

High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion

We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent under mild conditions of the involved amplitude function and for some interesting cases the coefficients can be rapidly computed, thus providing a viable alternative to the conventional dichotomy between series expansion and asymptotic expansion. The present method has been extensively tested in different regimes of the parameters and compared with recently investigated convergent and uniform asymptotic expansions.     

Numerical methods for the computation of the confluent and Gauss hypergeometric functions

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.     

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a,b and z

In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a～b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.     

Properties of the zeros of confluent hypergeometric functions

Several infinite systems of nonlinear algebraic equations satisfied by the zeros of confluent hypergeometric functions are derived. Certain sum rules and other related properties for the zeros follow from these equations. A large class of special functions, which are special cases of confluent hypergeometric functions, is included. This is illustrated in the case of the zeros of Bessel functions and Laguerre polynomials.     

On the functional properties of the confluent hypergeometric zeta-function

A zeta-function associated with Kummer’s confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan’s formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.     

On some solutions of the extended confluent hypergeometric differential equation

The extended confluent hypergeometric equation is defined (Section 1) as a linear second-order differential equation with (Section 2) a regular singularity at the origin and an (Section 3) irregular singularity of arbitrary degree M+1 at infinity; the original confluent hypergeometric equation is the particular case M=0, whereas the case M=1 is reducible (Section 3.2) to the former. Six types of solutions of the extended confluent hypergeometric equation of degree M, are obtained viz.: (i) functions of the first kind, i.e., regular ascending power series expansions, with infinite radius of convergence about the origin, for all values of the coefficients and degree M (Section 2.1); (ii) functions of the second kind, i.e. power series expansions with a logarithmic singularity, at the origin, for some values of the coefficients and all M (Section 2.2): (iii) only one asymptotic power series expansion exists, (Section 2.3) for M=0; (iv) concerning normal integrals (Section 3.2), valid as asymptotic expansions in the neighbourhood of the point-at-infinity, one exists for degree zero M=0 and two for degree unity M=1; (v) for degree greater than one M>1, two Laurent series expansions (Section 3.3) valid in the neighbourhood of infinity are obtained; (vi) an integral representation (Section 4) using the complex Laplace transform (Section 4.1) is obtained for (Sections 4.2–4.3) degree unity or zero M⩽1, using paths in a complex cut-plane (Fig. 1).     

Convergence of Magnus integral addition theorems for confluent hypergeometric functions

In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind U with argument x+y expressed as an integral of a product of two U's, one with argument x and another with argument y. We take advantage of recently obtained asymptotics for U with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of U, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.     

On the expansion of confluent hypergeometric functions in terms of Bessel functions

For the confluent hypergeometric functions U (a, b, z) and M (a, b, z) asymptotic expansions are given for a → ∞. The expansions contain modified Bessel functions. For real values of the parameters rigorous error bounds are given.     

关于φ（z）f（z）f＇（z）的值分布

本文研究有穷圆内的值分布，其中ｆ（２）为非常数亚纯函数，为非零亚纯函数，并对平面上这样的函数，若，得到     

Convergence of denominators of joint padé approximations of a set of confluent hypergeometric functions

Translated from Matematicheskii Zhurnal, Vol. 40, No. 6, pp. 792–795, November–December, 1988.