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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


On bounds for Kummer’s function ratio
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by Lukas Sablica and Kurt Hornik HTML | PDF
Math. Comp. 91 (2022), 887-907 Request permission


In this paper we present lower and upper bounds for Kummer’s function ratios of the form $\frac {{M(a, b, z)}’}{M(a, b, z)}$ when $0<a<b$. 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.
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Additional Information
  • Lukas Sablica
  • Affiliation: Institute for Statistics and Mathematics, Vienna University of Economics and Business, Austria
  • ORCID: 0000-0001-9166-4563
  • Email:
  • Kurt Hornik
  • Affiliation: Institute for Statistics and Mathematics, Vienna University of Economics and Business, Austria
  • MR Author ID: 265899
  • ORCID: 0000-0003-4198-9911
  • Email:
  • Received by editor(s): May 18, 2020
  • Received by editor(s) in revised form: March 22, 2021, and July 23, 2021
  • Published electronically: November 5, 2021
  • © Copyright 2009 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 887-907
  • MSC (2020): Primary 33F05, 33C15; Secondary 65B99, 65D15
  • DOI:
  • MathSciNet review: 4379980