A Conjectured Integer Sequence Arising From the Exponential Integral

Richard P. Brent; M. L. Glasser; Anthony J. Guttmann

paper

A Conjectured Integer Sequence Arising From the Exponential Integral

arXiv:1812.00316

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 $ρ_{n} = a_{n} b_{n}$ , so $\sum ρ_{n} z^{n} = (f_{0} ⊙ f_{1}) (z)$ is a Hadamard product. We obtain an asymptotic expansion $2 n^{3/2} ρ_{n} \sim - \sum d_{k} n^{- k}$ as $n \to \infty$ , where the $d_{k} \in Q$ , $d_{0} = 1$ . We conjecture that $2^{6 k} d_{k} \in Z$ . This has been verified for $k \leq 1000$ .

18 pages, additional motivation and references in v3/v4