A Variation of the Morris Constant Term

The 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 hₙ(t), in the study of the Ehrhart polynomial Hₙ(t) of the n-th Birkhoff polytope, which consists of all doubly stochastic matrices of order n. The constant term hₙ(t) corresponds to a particular constant term in the study of Hₙ(t). We give a characterization of hₙ(t) as a polynomial of degree (n-1)² with additional nice properties involving the Morris constant term. We also construct a recursion for hₙ(t) using a technique that is similar to the one used by Baldoni-Silva and Vergne, and by Xin in their proofs of the Morris constant term identity. Using this method, we obtain explicit formulas of hₙ(t) for 3 ≤ n ≤ 29 without difficulty.