Hardy's generalization of eᶻ and related analogs of cosine and sine

Abstract

In 1904, Hardy introduced an entire function depending on two parameters being a generalization of e z. He had studied in detail its asymptotic properties and that of its zeros. We consider the two following non-asymptotic problems related to the zeros. (i) Determine values of the parameters such that all the zeros belong to the open left half-plane. For these values, the analogs of sine and cosine generated by Hardy’s function have real, simple and interlacing zeros. (ii) Determine the number of real zeros as a function of the parameters.