Stieltjes moment sequences of polynomials

A sequence $(a_n)_{n geq 0}$ is Stieltjes moment sequence if it has the form $a_n = int_0^infty x^n dmu(x)$ for $mu$ is a nonnegative measure on $[0,infty)$. It is known that $(a_n)_{n geq 0}$ is a Stieltjes moment sequence if and only if the matrix $H =[a_{i+j}]_{i,j geq 0}$ is totally positive, i.e., all its minors are nonnegative. We define a sequence of polynomials in $x_1,x_2,ldots,x_n$ $(a_n(x_1,x_2,ldots,x_n))_{n geq 0}$ to be a Stieltjes moment sequence of polynomials if the matrix $H =[a_{i+j} (x_1,x_2,ldots,x_n)]_{i,j geq 0}$ is $(x_1,x_2,ldots,x_n)$-totally positive, i.e., all its minors are polynomials in $x_1,x_2,ldots,x_n$ 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.