Evaluations of determinants of several matrices whose entries satisfy inhomogeneous recurrence relations

Abstract

Let P = [Pᵢ,ⱼ]ᵢ,ⱼ ≥ 0 be an infinite matrix whose entries satisfy the recurrence relation Pᵢ,ⱼ = xPᵢ,ⱼ₋₁ + yPᵢ₋₁,ⱼ₋₁ + zPᵢ₋₁,ⱼ + f(i, j) for i, j ≥ 1, where x, y, and z are arbitrary constants, and f is a function mapping ℕ × ℕ to ℂ. The purpose of this article is twofold: First, it provides a factorization of P for specific values of x, y, and z, as well as the function f. Second, it presents several evaluations of the determinants of such matrices.