On Two Sequences of Orthogonal Polynomials Related to Jordan Blocks

Abstract

We study two infinite sequences of polynomials related to Jordan blocks that have various interesting properties. We show that they are orthogonal polynomials whose sequences of moments are Catalan numbers and we relate them explicitly to the Chebyshev polynomials. We also use them to compute the singular values of some Jordan blocks. Finally, we investigate some combinatorial properties of the inverse sequences of these polynomials; we show them to be intimately related to the convolutions of the Catalan sequence.