On d-orthogonal Tchebychev polynomials, I

Abstract

Let { P n } n≥0 be a sequence of monic polynomials, { u n } n≥0 the dual sequence associated with { P n } n≥0 . Now let Q n(x) ≔ P′ n+1(x) (n + 1) , n ≥ 0, and { P n (1)} n≥0 the associated polynomials of { P n } n≥0 . We consider the following two problems: 1. (1) Find all d-orthogonal polynomial sequences { P n } n≥0 which satisfy P n (1) = P n , n = 0, 1, 2,…. 2. (2) Find all d-orthogonal polynomial sequences { P n } n≥0 which satisfy P n (1) = Q n , n = 0, 1, 2, …. Remember that a d-orthogonal polynomial sequence ( d-OPS) is defined by systems of orthogonality relations or, equivalently, a polynomial sequence satisfying a ( d+1)-order recurrence relation (see Definition 1.1). In this paper we consider only the problem (1). The resulting polynomials are natural extensions of the Tchebychev monic polynomials of the second kind. Their recurrence coeffcients and generating function are explicitly determined. In particular, when d = 2, we obtain that any polynomial satisfies a third-order differential equation and under certain assumptions on the recurrence coefficients, we give integral representations of the linear functionals with respect to which the polynomials are 2-orthogonal. The analogous problem (2) will be studied in a separate paper.