In this paper, we give some new explicit relations between two families of polynomials defined by recurrence relations of all order. These relations allow us to analyze, even in the Sobolev case, how some properties of a family of orthogonal polynomials are affected when the coefficients of the recurrence relation and the order are perturbed. In a paper we have already given a method which allows us to study the polynomials defined by a threeterm recurrence relation. Also here some generalizations are given.Keywords Perturbed orthogonal polynomials system · General recurrence relations with perturbed coefficients · Sobolev recurrence relations · Differential equations · Zeros

Abstract. We show that polynomials defined by recurrence relations with periodic coefficients may be represented with the help of Chebyshev polynomials of the second kind.Introduction. The class of orthogonal polynomials studied in this paper served as a starting point for several authors [1-4] when studying more general classes of orthogonal polynomials or continued fractions. The aim of this note is to show that, on the other hand, this class can be described with the help of the classical Chebyshev polynomials.Let {P n } n , deg P n = n, be a sequence of polynomials defined by a threeterm recurrence

11 pages; latex2e; pas de figuresThe orthogonal polynomials with recurrence relation \[(\la_n+\mu_n-z)\,F_n(z)=\mu_{n+1}\,F_{n+1}(z)+\la_{n-1}\,F_{n-1}(z)\] and the three kinds of cubic transition rates \[\left\{\barr{ll} \la_n=(3n+1)^2(3n+2), & \qq\mu_n=(3n-1)(3n)^2,\\[4mm] \la_n=(3n+2)^2(3n+3), & \qq\mu_n=3n(3n+1)^2,\\[4mm] \la_n=(3n+1)(3n+2)^2, & \qq\mu_n=(3n)^2(3n+1),\earr\right.\] correspond to indeterminate Stieltjes moment problems. It follows that the polynomials $\,F_n(z)\,$ have infinitely many orthogonality measures, whose Stieltjes transform is obtained from their Nevanlinna matrix, a $2\times 2$ matrix of entire functions. We present the full Nevanlinna matrix for these three classes of polynomials and we discuss its growthat infinity and the asymptotic behaviour of the spectra of the Nevanlinna extremal measures

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