AN ASYMPTOTIC EXPANSION INSPIRED BY RAMANUJAN

RICHARD P. BRENT

Abstract

Corollary 2, Entry 9, Chapter 4 of Ramanujan’s first notebook [1] claims that

∑_{k=1}^{∞} \frac{(−1)^{k−1}}{nk} (\frac{x^{k}}{k!} )^{n} ∼ ln x + γ

as x → ∞. This is known to be correct for the case n = 1, but incorrect for n ≥ 3. We show that the result is correct for n = 2. We also consider the order of the error term, and discuss a different, correct generalisation of the case n = 1.

Comments

Only the Abstract is given here. The full report appeared as [2]. The results were also presented and appeared in [3]. The generalisation mentioned in the Abstract was suggested by Brent and McMillan in [4].

References

[1] B. C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.

[2] R. P. Brent An asymptotic expansion inspired by Ramanujan, Technical Report CMA-MR02-93/SMS-10-93, CMA, ANU, February 1993, 7 pp. Also appeared in Australian Math. Soc. Gazette 20 (1993), 149-155. rpb139tr.

[3] R. P. Brent “Ramanujan and Euler’s constant”, presented at Mathematics of Computation 1943–1993, Vancouver, 9–13 August 1993. Appeared in Proceedings of Symposia in Applied Mathematics 48 (1994), 541-545. rpb139.

[4] R. P. Brent and E. M. McMillan, “Some new algorithms for high-precision computation of Euler’s constant”, Math. Comp. 34 (1980), 305–312. MR 82g:10002. rpb049.

Computer Sciences Laboratory, Australian National University, Canberra, ACT 0200

   E-mail address: rpb@cslab.anu.edu.au

1991 Mathematics Subject Classification. Primary 01A60, 33C10, 41A60; Secondary 33-01, 33-03, 33E99, 40A25, 65D20.

Key words and phrases. Ramanujan, Euler’s constant, Bessel functions, asymptotic expansion, exponential integral, harmonic number.

Copyright \begin{matrix}c \\ ◯ \end{matrix} 1993, R. P. Brent. rpb139a typeset using AMS-L^{A}TEX.