On Kaluza's sign criterion for reciprocal power series (Q2890377)

Let \(f(x) = \sum_{n=0}^\infty a_n x^n\) be a Maclaurin series with positive radius of convergence and with positive \(a_n\). Kaluza proved in 1928 that for \(\log\)-convex \((a_n)\) the reciprocal series NEWLINE\[NEWLINE \frac{1}{f(x)} = \frac{1}{a_0} + \sum_{n=1}^\infty b_n x^n NEWLINE\]NEWLINE has non-positive coefficients \(b_n\) (\(n\in\mathbb N\)). In this paper, the sharpness of the \(\log\)-convexity condition is explored and various examples in terms of the Gaussian hypergeometric series are given.