Representing polynomials as characteristic polynomials via the Stieltjes transform

The Stieltjes transform SA of an infinite lower triangular matrix A with nonzero diagonal entries is defined by SA=A−1A¯ where A¯ is the matrix obtained from A by deleting its initial row. In this paper, we express a sequence of polynomials as the characteristic polynomials of the Stieltjes transforms using a highly structured infinite lower triangular matrix called a Riordan matrix. As a result, computation of the zeros of such polynomials becomes amenable to iterative methods for computing eigenvalues, or to eigenvalue location theorems such as the Geršgorin theorem. We also describe a finite analog of the polynomial correspondence and its relationship to eigenvalue regions. As an application, the recurrence relations for several polynomial sequences are obtained using the Stieltjes transform.