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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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

Abstract:

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: lukas.sablica@wu.ac.at
  • 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: kurt.hornik@wu.ac.at
  • 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: https://doi.org/10.1090/mcom/3690
  • MathSciNet review: 4379980