On Kaluza's sign criterion for reciprocal power series

TL;DR

This paper examines Kaluza's sign criterion for reciprocal power series, exploring its sharpness through examples involving hypergeometric series and applying related monotonicity criteria.

Contribution

It analyzes the precision of Kaluza's criterion and demonstrates its application using hypergeometric series and monotonicity conditions.

Findings

Kaluza's criterion is sharp in certain cases

Examples with hypergeometric series illustrate the criterion

Monotonicity of power series quotients is connected to the criterion

Abstract

T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzy\.z is applied.