## On bounds for Kummer’s function ratio

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Lukas Sablica and Kurt Hornik
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## 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.## References

- Rémy Abergel and Lionel Moisan. 2016.
*Fast and accurate evaluation of a generalized incomplete gamma function*. - D. E. Amos,
*Computation of modified Bessel functions and their ratios*, Math. Comp.**28**(1974), 239–251. MR**333287**, DOI 10.1090/S0025-5718-1974-0333287-7 - Avleen S. Bijral, Markus Breitenbach, and Greg Grudic. 2007.
*Mixture of Watson distributions: A generative model for hyperspherical embeddings*. In Marina Meila and Xiaotong Shen, eds, AISTATS 2007: Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, volume 2 of JMLR Workshop and Conference Proceedings, pp. 35–42, San Juan, Puerto Rico, http://jmlr.org/proceedings/papers/v2/bijral07a/bijral07a.pdf. - Phelim Boyle and Alex Potapchik,
*Application of high-precision computing for pricing arithmetic Asian options*, ISSAC 2006, ACM, New York, 2006, pp. 39–46. MR**2289099**, DOI 10.1145/1145768.1145782 - Blanca Bujanda, José L. López, and Pedro J. Pagola,
*Convergent expansions of the confluent hypergeometric functions in terms of elementary functions*, Math. Comp.**88**(2019), no. 318, 1773–1789. MR**3925484**, DOI 10.1090/mcom/3389 - DLMF.
*NIST Digital library of mathematical functions*. http://dlmf.nist.gov/, Release 1.0.19 of 2018-06-22, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds. http://dlmf.nist.gov/. - V. Eremenko, N. J. Upadhyay, I. J. Thompson, Ch Elster, F. M. Nunes, G. Arbanas, J. E. Escher, and L. Hlophe. 2015.
*Coulomb wave functions in momentum space*. Comput. Phys. Commun.**88**, no. 2 195–203 DOI:10.1016/j.cpc.2014.10.002. - Walter Gautschi,
*A computational procedure for incomplete gamma functions*, Rend. Sem. Mat. Univ. Politec. Torino**37**(1979), no. 1, 1–9 (Italian). MR**547763** - Walter Gautschi,
*Anomalous convergence of a continued fraction for ratios of Kummer functions*, Math. Comp.**31**(1977), no. 140, 994–999. MR**442204**, DOI 10.1090/S0025-5718-1977-0442204-3 - Walter Gautschi and Josef Slavik,
*On the computation of modified Bessel function ratios*, Math. Comp.**32**(1978), no. 143, 865–875. MR**470267**, DOI 10.1090/S0025-5718-1978-0470267-9 - Mihai Gavrila. 1967.
*Elastic scattering of photons by a hydrogen atom*. Phys. Rev.**163**, 147–155 DOI: 10.1103/PhysRev.163.147. J. Stat. Soft. - Kurt Hornik and Bettina Grün,
*Amos-type bounds for modified Bessel function ratios*, J. Math. Anal. Appl.**408**(2013), no. 1, 91–101. MR**3079949**, DOI 10.1016/j.jmaa.2013.05.070 - Kurt Hornik and Bettina Grün. 2014.
*movMF: An R package for fitting mixtures of von Mises-Fisher distributions*.*Journal of Statistical Software*,**58**, no. 10, 1–31. DOI: 10.18637/jss.v058.i10. URL https://www.jstatsoft.org/v058/i10. - S. I. Kalmykov and D. B. Karp,
*Log-concavity for series in reciprocal gamma functions and applications*, Integral Transforms Spec. Funct.**24**(2013), no. 11, 859–872. MR**3171999**, DOI 10.1080/10652469.2013.764874 - D. Karp and S. M. Sitnik,
*Inequalities and monotonicity of ratios for generalized hypergeometric function*, J. Approx. Theory**161**(2009), no. 1, 337–352. MR**2558159**, DOI 10.1016/j.jat.2008.10.002 - Lisa Lorentzen and Haakon Waadeland,
*Continued fractions. Vol. 1*, 2nd ed., Atlantis Studies in Mathematics for Engineering and Science, vol. 1, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. Convergence theory. MR**2433845**, DOI 10.2991/978-94-91216-37-4 - Keith E. Muller,
*Computing the confluent hypergeometric function, $M(a,b,x)$*, Numer. Math.**90**(2001), no. 1, 179–196. MR**1868767**, DOI 10.1007/s002110100285 - Oskar Perron,
*Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche*, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957 (German). MR**0085349** - Javier Segura,
*Bounds for ratios of modified Bessel functions and associated Turán-type inequalities*, J. Math. Anal. Appl.**374**(2011), no. 2, 516–528. MR**2729238**, DOI 10.1016/j.jmaa.2010.09.030 - Javier Segura,
*On bounds for solutions of monotonic first order difference-differential systems*, J. Inequal. Appl. , posted on (2012), 2012:65, 17. MR**2915630**, DOI 10.1186/1029-242X-2012-65 - Javier Segura,
*Sharp bounds for cumulative distribution functions*, J. Math. Anal. Appl.**436**(2016), no. 2, 748–763. MR**3446977**, DOI 10.1016/j.jmaa.2015.12.024 - Suvrit Sra and Dmitrii Karp,
*The multivariate Watson distribution: maximum-likelihood estimation and other aspects*, J. Multivariate Anal.**114**(2013), 256–269. MR**2993885**, DOI 10.1016/j.jmva.2012.08.010 - Wolfram Research, Inc. 2019.
*Mathematica, Version 12.0*. Champaign, IL, https://www.wolfram.com/mathematica.

## 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