Stieltjes moment sequences of polynomials

A sequence $(a_n)_{n \geq 0}$ is Stieltjes moment sequence if it has the form $a_n = \int_0^\infty x^n d\mu(x)$ 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.