Moment functions of higher rank on polynomial hypergroups

TL;DR

This paper extends the study of generalized moment functions of higher order from commutative groups to polynomial hypergroups, providing characterizations and explicit formulas, including for Tchebyshev hypergroups.

Contribution

It characterizes higher order moment functions on polynomial hypergroups using derivatives of polynomial compositions, advancing the understanding beyond commutative groups.

Findings

Characterization of moment functions on polynomial hypergroups.

Explicit formulas for rank at most two on Tchebyshev hypergroups.

Extension of previous group-based results to hypergroup structures.

Abstract

In this paper we consider generalized moment functions of higher order. These functions are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. In our former paper we investigated the properties of generalized moment functions of higher order on commutative groups. In particular, we proved the characterization of generalized moment functions on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and a sequence additive functions. In the present paper we continue the study of generalized moment function sequences of higher order in the more abstract setting, namely we consider functions defined on a hypergroup. We characterize these functions on the polynomial hypergroup in one variable by means of partial derivatives of a composition of polynomials generating the polynomial hypergroup and an analytic function. As an example, we give an explicit formula for moment generating functions of rank at most two on the Tchebyshev hypergroup.