On the Pochhammer transformation and hyperbolic polynomials decomposed in the Pochhammer basis

Abstract

For a given we consider the linear operator on the space of complex polynomials given by the formula We call the h-Pochhammer transformation. In this paper we investigate the properties of the Pochhammer transformation, in particular, how it affects the first and last roots of a hyperbolic polynomial. We study how the coefficients in the decomposition of a polynomial with respect to the Pochhammer basis are related to its roots. We also prove an h-analogue of the famous Schur Convolution Theorem.