Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters

ABSTRACT

New asymptotic expansions are derived of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of a and b, with z fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case $a\sim b$. We use a uniform method to handle all cases of these parameters.

Keywords:

• Asymptotic expansions
• Kummer functions
• confluent hypergeometric functions

2010 Mathematics Subject Classifications:

• Primary 41A60
• Secondary 33C15

Acknowledgments

The author is grateful to the reviewer for careful reading earlier versions of the manuscript and for helpful comments that improved the article. The author thanks CWI, Amsterdam, for scientific support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This follows from the first functions given in (A14(A14) $R_1(\sigma ,s,\mu )=\frac{s}{(\sigma -\mu )(\sigma -s{)}^2},R_2(\sigma ,s,\mu )=\frac{s(\mu s+\mu \sigma -2{\sigma }^2)}{(\sigma -\mu {)}^3(\sigma -s{)}^3}.$(A14) ) and induction with respect to n.

2 The reviewer observed: It seems that the numerical coefficients are the same as the sequence A269940 in the OEIS. It would be worth investigating this in the future. See also https://oeis.org/A269940.

Additional information

Funding

This work was supported by the Spanish Ministerio de Ciencia, Innovación y Universidades under Grants MTM2015-67142-P (MINECO/FEDER, UE) and PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE).