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On the asymptotic behaviour of some sequences of polynomials generated by second order linear recurrence relations. (Spanish. English summary) Zbl 0973.42014
Summary: The asymptotic behaviour of systems of polynomials defined by linear second-order recurrence relations of the form \[ S_{n+1}(z)- (a_n\cdot z+ b_n)\cdot S_n(z)+ c_n\cdot S_{n-1}(z)= 0, \] where \(a_n\to 2\), \(b_n\to 0\), \(c_n\to 1\), is studied. As an application of the results, the domains of convergence of polynomial series of the form \[ \sum^\infty_{n=0} p_nS_n(x),\qquad p_n\in\mathbb{C}, \] with \(\limsup\sqrt{|p_n|}= {1\over\rho}< 1\) are determined, and the well-known Abel’s lemma for power series is extended to such series under appropriate assumptions. The latter problem had been previously addressed by M. A. Dumett [Lect. Mat. 16, No. 1, 37-61 (1995; Zbl 0868.33004)] from a different point of view.
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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