Abstract. We introduce a family of weight matrices W of the form T (t)T * (t), T (t) = e A t e Dt 2 , where A is certain nilpotent matrix and D is a diagonal matrix with negative real entries. The weight matrices W have arbitrary size N × N and depend on N parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second order differential equation with differential coefficients that are matrix polynomials F 2 , F 1 and F 0 (independent of n) of degrees not bigger than 2, 1 and 0 respectively. For size 2 × 2, we find an explicit expression for a sequence of orthonormal polynomials with respect to W . In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.

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An action is a continuous function α: T × X → X, where T is a semigroup, X is a Hausdorff space, and α(t1, α(t2, x)) = α(t1,t2x) for all t1, t2 ∈ T and x ∈ X . If, for an action α, Q(α) = {x ∈ X| α(T × {x}) = X} is non-empty, then α is called a point-transitive action. Our aim in this note is to classify the point-transitive actions of the unit interval with the usual, nil, or min multiplications.The reader is referred to [5; 7; 9] for information concerning the general theory of semigroups. All semigroups which are considered here are compact and Abelian and all spaces are compact Hausdorff. Actions by semigroups have been studied in [1; 3; 8].

Let S and T be compact (topological) semigroups, A be a closed subsemigroup of S, and f be a continuous homomorphism of A onto T. It is a natural question to ask is there a compact semigroup Z containing T and a continuous homomorphism φ of S onto Z such that φ restricted to A is f?In the category of compact Hausdorff spaces, the answer to the analogous question may be given by the following construction due to Borsuk. Let X and Y be compact Hausdorff spaces, A be a closed subspace of X and f a continuous function of A onto Y. Then R(f, X) = {(x, y)|f(x) = f(y) or x = y} is a closed equivalence relation on X. The adjunction space Z(f, x) is X/R(f, X). The purpose of this note is to investigate this construction in semigroups.