A Conjectured Integer Sequence Arising From the Exponential Integral

Abstract

Let f0(z)=exp(z/(1−z)), f1(z)=exp(1/(1−z))E1(1/(1−z)), where E1(x)=∫∞xe−tt−1dt. Let an=[zn]f0(z) and bn=[zn]f1(z) be the corresponding Maclaurin series coefficients. We show that an and bn may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences (an) and (bn) as n→∞, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (bn). Let ρn=anbn, so ∑ρnzn=(f0⊙f1)(z) is a Hadamard product. We obtain an asymptotic expansion 2n3/2ρn∼−∑dkn−k as n→∞, where the dk∈Q, d0=1. We conjecture that 26kdk∈Z. This has been verified for k≤1000.