Formal series and an algorithm for computing some special determinants with elements in a ring

Abstract

We define an algorithm for computing the determinant of an n×n upper Hessenberg matrix A=[a ij ]. When the algorithm is specialized, it yields an algorithm for computing the determinant of an n×n Toeplitz matrix with a ij =0, i>j+1, where a ij is a member of a noncommutative ring, A. Such determinants turn out to play an interesting role concerning solutions of certain operator equations in the algebra of formal power series with coefficients in the ring A. By specializing the algorithm we show how certain moment integrals may be computed