On bounds for Kummer’s function ratio

Abstract

In this paper we present lower and upper bounds for Kummer’s function ratios of the form M ( a , b , z ) ′ M ( a , b , z ) \frac {{M(a, b, z)}’}{M(a, b, z)} when 0 > a > b 0>a>b . The derived bounds are asymptotically precise, theoretically well-defined, numerically accurate, and easy to compute. Moreover, we show how the bounds can be used as starting values for monotonically convergent sequences to approximate the ratio with even higher precision while avoiding the anomalous convergence discussed by Gautschi [Math. Comp. 31 (1977), pp. 994–999]. This allows to apply the results in multiple areas, as for example the estimation of Watson distributions in statistical modelling. Furthermore, we extend the convergence results provided by Gautschi and the list of known bounds for the inverse of Kummer’s function ratio given by Sra and Karp [J. Multivariate Anal. 114 (2013), pp. 256–269]. In addition, the derived starting bounds are compared and connected to other results from the literature.