Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions

TL;DR

This paper analyzes the non-asymptotic zeros of Whittaker and Kummer functions, correcting previous misconceptions and providing new estimates on their zero locations in the complex plane.

Contribution

It introduces a corrected characterization of zeros' locations for these functions, based on Hille's technique, improving upon previous literature.

Findings

Characterization of the sign of the real part of zeros

Estimates of zero regions in the complex plane

Correction of Tsvetkov's previous statement

Abstract

This paper focuses on the location of the non-asymptotic zeros of Whittaker and Kummer confluent hypergeometric functions. Based on a technique by E. Hille for the analysis of solutions of some second-order ordinary differential equations, we characterize the sign of the real part of zeros of Whittaker and Kummer functions and provide estimates on the regions of the complex plane where those zeros can be located. Our main result is a correction of a previous statement by G. E. Tsvetkov whose propagation has induced mistakes in the literature. In particular, we review some results of E. B. Saff and R. S. Varga on the error of Pad{\'e}'s rational approximation of the exponential function, which are based on the latter.