A Conjectured Integer Sequence Arising From the Exponential Integral

Let $f_0(z) = \exp(z/(1-z))$ , $f_1(z) = \exp(1/(1-z))E_1(1/(1-z))$ , where $E_1(x) = \int_x^\infty e^{-t}t^{-1}{\,d}t$ . Let a_n = [zⁿ]f₀(z) and b_n = [zⁿ]f₁(z) be the corresponding Maclaurin series coefficients. We show that a_n and b_n may be expressed in terms of confluent hypergeometric functions.

We consider the asymptotic behaviour of the sequences (a_n) and (b_n) as $n \to \infty$ , showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (b_n).

Let $\rho_n = a_n b_n$ , so $\sum \rho_n z^n = (f_0{\,\odot}f_1)(z)$ is a Hadamard product. We obtain an asymptotic expansion $2n^{3/2}\rho_n \sim -\sum d_k n^{-k}$ as $n \to \infty$ , where $d_k\in{\mathbb Q}$ , d₀=1. We conjecture that $2^{6k}d_k \in {\mathbb Z}$ . This has been verified for $k \le 1000$ .

A Conjectured Integer Sequence Arising From the Exponential Integral

Richard P. Brent Australian National University Canberra ACT 2601 Australia M. L. Glasser Clarkson University Potsdam, NY 13699-5820 USA Anthony J. Guttmann School of Mathematics and Statistics The University of Melbourne Vic. 3010 Australia

Richard P. Brent
Australian National University
Canberra
ACT 2601
Australia
M. L. Glasser
Clarkson University
Potsdam, NY 13699-5820
USA
Anthony J. Guttmann
School of Mathematics and Statistics
The University of Melbourne
Vic. 3010
Australia