Abstract
We give some structural formulas for the family of matrix-valued orthogonal polynomials of size 2times 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.
Highlights
In the last few years, the search for examples of matrix-valued orthogonal polynomials that are common eigenfunctions of a second-order differential operator, that is to say, satisfying a bispectral property in the sense of [13], has received a lot of attention after the seminal work of A
The theory of matrix-valued orthogonal polynomials was started by Krein in 1949 [37,38], in connection with spectral analysis and moment problems
For a given weight matrix W, the analysis of the algebra D(W ) of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to W as eigenfunctions has received much attention in the literature in the last fifteen years [6,8,9,33,42,45,47]
Summary
In the last few years, the search for examples of matrix-valued orthogonal polynomials that are common eigenfunctions of a second-order differential operator, that is to say, satisfying a bispectral property in the sense of [13], has received a lot of attention after the seminal work of A. The polynomials (Pn(α,β,v))n≥0 introduced in [4], orthogonal with respect to the weight matrix W (α,β,v) given in (2.4) and (2.5), are common eigenfunctions of an hypergeometric operator with matrix eigenvalues n, which are diagonal matrices with no repetition in their entries This fact could be especially useful if one intends to use this family of polynomials in the context of time and band limiting, where the commutativity of the matrix-valued eigenvalues ( n)n could play an important role. For a given weight matrix W , the analysis of the algebra D(W ) of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to W as eigenfunctions has received much attention in the literature in the last fifteen years [6,8,9,33,42,45,47]. While for classical orthogonal polynomials, the structure of this algebra is very well-known (see [39]), in the matrix setting, where this algebra is non-commutative, the situation is highly non-trivial
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