A Conjectured Integer Sequence Arising From the Exponential Integral

Abstract

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 $a_n = [z^n]f_0(z)$ and $b_n = [z^n]f_1(z)$ 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 the $d_k\in\mathbb Q$, $d_0=1$. We conjecture that $2^{6k}d_k \in \mathbb Z$. This has been verified for $k \le 1000$.