We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν > 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν + 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials Approx (2017) 46:459-487 are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauertype polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν = 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.

We develop a method that allows us to construct families of orthogonal matrix polynomials of size N × N satisfying second order difference equations with polynomial coefficients. The existence (and properties) of these orthogonal families strongly depends on the non commutativity of the matrix product, the existence of singular matrices and the matrix size N.Ministerio de Economía y CompetitividadJunta de AndalucíaFondo Europeo de Desarrollo Regiona

Matrix-valued analogues of the little q-Jacobi polynomials are introduced and studied. For the 2×2-matrix-valued little q-Jacobi polynomials explicit expressions for the orthogonality relations, Rodrigues formula, three-term recurrence relation and their relation to matrix-valued q-hypergeometric series and the scalar-valued little q-Jacobi polynomials are presented. The study is based on a matrix-valued q-difference operator, which is a q-analogue of Tirao's matrix-valued hypergeometric differential operator.

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.

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