Stieltjes moment sequences of polynomials

Abstract

A sequence $(a_n{)}_{n\ge 0}$ is Stieltjes moment sequence if it has the form $a_n={\int }_0^{\infty }{x}^{n}d\mu (x)$ for $\mu$ is a nonnegative measure on $[0,\infty )$. It is known that $(a_n{)}_{n\ge 0}$ is a Stieltjes moment sequence if and only if the matrix $H=[a_{i+j}{]}_{i,j\ge 0}$ is totally positive, i.e., all its minors are nonnegative. We define a sequence of polynomials in $x_1,x_2,\dots ,x_n$ $(a_n(x_1,x_2,\dots ,x_n){)}_{n\ge 0}$ to be a Stieltjes moment sequence of polynomials if the matrix $H=[a_{i+j}(x_1,x_2,\dots ,x_n){]}_{i,j\ge 0}$ is $(x_1,x_2,\dots ,x_n)$-totally positive, i.e., all its minors are polynomials in $x_1,x_2,\dots ,x_n$ with nonnegative coefficients. The main goal of this paper is to produce a large class of Stieltjes moment sequences of polynomials by finding multivariable analogues of Catalan-like numbers as defined by Aigner.