On Some Conjectures by Morris et al. about Zeros of an Entire Function

It is known that the entire function `ns0 yz.nqn ny1.r2rn!, qg 0, 1., has infinitely many positive but no other zeros in the complex plane. A number of conjectures on these zeros have been made by Morris, Feldstein, and Bowen and by Iserles. The main objective of this note is to establish some connection among these conjectures and to prove the following estimates: tnq1 1 tnq1 1 lim inf G , lim sup F , nª` tn q nª` tn q2 where t0, t1, . . . are the zeros in monotonic increasing order. Some identities involving those zeros are proved by using Hadamard’s factorisation theorem.