Efficient and Precise Calculation of the Confluent Hypergeometric Function

Abstract

Kummer's function, also known as the confluent hypergeometric function (CHF), is an important mathematical function, in particular due to its many special cases, which include the Bessel function, the incomplete Gamma function and the error function (erf). The CHF has no closed form expression, but instead is most commonly expressed as an infinite sum of ratios of rising factorials, which makes its precise and efficient calculation challenging. It is a function of three parameters, the first two being the rising factorial base of the numerator and denominator, and the third being a scale parameter. Accurate and efficient calculation for large values of the scale parameter is particularly challenging due to numeric underflow and overflow which easily occur when summing the underlying component terms. This work presents an elegant and precise mathematical algorithm for the calculation of the CHF, which is of particular advantage for large values of the scale parameter. This method massively reduces the number and range of component terms which need to be summed to achieve any required precision, thus obviating the need for the computationally intensive transformations needed by current algorithms.