Asymptotic behaviours and general recurrence relations

In this paper, we present some new results which give greater efficiency to our method – expanded in preceding papers, for studying some asymptotic behaviours of polynomials generated by recurrence relations. More precisely, we give some new results which show that, for the asymptotic of the polynomials satisfying recurrence relations, some interesting well-known studies in the literature can be extended to recurrence relations of other kinds. For example, in contrast with what is known, we explicitly show that there exist many families of polynomials, say { M k } and { N k } with M k a perturbation of N k such that lim k → ∞ M k ( z ) / N k ( z ) = 1 uniformly in a subset – which may be very large – of the complex plane and with conditions on perturbations which, to our knowledge, are new. We illustrate that by some surprising examples. This work also allows us to extend our analysis to other classes of orthogonal polynomials.