The convex cone of weight matrices associated to a second-order matrix difference operator

Abstract

We associate to a given finite order difference operator D with matrix coefficients the convex cone ϒ(D) formed by all weight matrices W with respect to which the operator D is symmetric. In the scalar case, the convex cone of positive measures associated to a second-order difference operator always reduces to the empty set except for those operators associated to the classical discrete families of Charlier, Meixner, Krawtchouk or Hahn, in which case the convex cone is the half line defined by the classical discrete measure itself. In the matrix case the situation is rather different. We develop two methods to study these convex cones and, using them, we construct some illustrative examples of second-order difference operators whose convex cones are, at least, two dimensional.