In the work presented below the classical subject of orthogonal polynomials on the real line is discussed in the matrix setting. An analogue of the determinant definition of orthogonal polynomials is presented; the classical properties such as the recurrence relation, the kernel polynomials, and the Christoffel-Darboux formula are discussed. A τ -function for the system of matrix-valued orthogonal polynomials on the real line is presented. Some properties of the τ -functions are investigated.

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It is well known that the four families of classical orthogonal polynomials (Jacobi, Bessel, Hermite and Laguerre) each satisfy an equation FP n (x) = λ n P n (x), n 0, for an appropriate second-order differential operator F. In this paper it is shown that any linear differential operator U which has the Jacobi, Bessel, Hermite or Laguerre polynomials as eigenfunctions has to be a polynomial with constant coefficients in the classical second-order operator F.

Abstract. In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel-Darboux formulas are presented for the first time.

Denote by K the operator of 'time-band-time' limiting on the surface of the sphere and consider the problem of computing singular vectors of K. This problem can be reduced to a simpler task of computing eigenfunctions of a differential operator, if a differential operator, which commutes with K and has a simple spectrum, can be exhibited. In Grünbaum et al (1982 SIAM J. Appl. Math. 42 941-55) such a second-order differential operator commuting with K on the appropriate subspaces was constructed. In this paper, this algebraic property of commutativity is used to produce an efficient numerical scheme for computing a convenient basis for the space of singular vectors of K. The basis forms an extended Chebyshev system, and a generalized Gaussian quadrature rule for such a basis is presented.