Hausdorff Moment Sequences Induced by Rational Functions

Abstract

We study the Hausdorff moment problem for a class of sequences, namely $(r(n))_{n\in {\mathbb {Z}}_+},$ where r is a rational function in the complex plane. We obtain a necessary condition for such sequence to be a Hausdorff moment sequence. We found an interesting connection between Hausdorff moment problem for this class of sequences with finite divided differences and convolution of complex exponential functions. We provide a sufficient condition on the zeros and poles of a rational function r so that $(r(n))_{n\in {\mathbb {Z}}_+}$ is a Hausdorff moment sequence. G. Misra asked whether the module tensor product of a subnormal module with the Hardy module over the polynomial ring is again a subnormal module or not. Using our necessary condition we answer the question of G. Misra in negative. Finally, we obtain a characterization of all real polynomials p of degree up to 4 and a certain class of real polynomials of degree 5 for which the sequence $(1/p(n))_{n\in {\mathbb {Z}}_+}$ is a Hausdorff moment sequence.