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 $b>a>0$, 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 $b>a>0$, 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.