A variation of the Morris constant term identity

TL;DR

This paper introduces a new variation of the Morris constant term identity, denoted $h_n(t)$, related to Ehrhart polynomials of Birkhoff polytopes, providing explicit formulas and recursive constructions for $n$ up to 29.

Contribution

It defines and characterizes a new polynomial variation $h_n(t)$ of the Morris constant term, with explicit formulas and recursive methods for computation.

Findings

Defined $h_n(t)$ as a polynomial of degree $(n-1)^2$ with special properties.

Constructed a recursion for $h_n(t)$ applicable for all integers $n \\geq 3$.

Produced explicit formulas for $h_n(t)$ for $3 \\leq n \\leq 29$.

Abstract

Morris constant term identity is important due to its equivalence with the well-known Selberg integral. We find a variation of the Morris constant term, denoted $h_n(t)$, in the study of the Ehrhart polynomial $H_n(t)$ of the $n$-th Birkhoff polytope, which consists of all doubly stochastic matrices of order $n$. The constant term $h_n(t)$ corresponds to a particular constant term in the study of $H_n(t)$. We give a characterization of $h_n(t)$ as a polynomial of degree $(n-1)^2$ with additional nice properties involving the Morris constant term identity. We also construct a recursion of $h_n(t)$ using a similar technique for the proof of the Morris constant term identity by Baldoni-Silva and Vergne, and by Xin. This method is applicable to any integer $n\geq 3$. We have produced explicit formulas of $h_n(t)$ for $3 \le n \le 29$ without difficulty.