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Abstract

Let f(x)=a d x d+a d−1 x d−1+⋅⋅⋅+a 0∊ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a d,a d−1,…,a 0) or (a d−1,a d−2,…,a 1) is close enough, in the l 1-distance, to the constant vector (b,b,…,b)∊ℝd+1 or ℝd−1, then all of its zeros have moduli 1.

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Abstract

We give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multiplier sequences for every simple set Q. We characterize sequences of real numbers which are multiplier sequences for every simple set Q, and obtain some results toward the partitioning of the set of classical multiplier sequences.

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Abstract

We derive sufficient conditions under which all but two zeros of reciprocal polynomials lie on the unit circle, and specify the location of the remaining two zeros.

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Abstract

We consider sequences of functions that have in some sense a universal distribution of limit points of zeros in the complex plane. In particular, we prove that functions having universal approximation properties on compact sets with connected complement automatically have such a universal distribution of limit points. Moreover, in the case of sequences of derivatives, we show connections between this kind of universality and some rather old results of Edrei/MacLane and Pólya. Finally, we show the lineability of the set what we call Jentzsch-universal power series.

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Abstract  

Recently Totik and Varjú [17] presented an estimate of the uniform norm of a monic polynomial with prescribed zeros on the unit circle. In this paper we improve their estimate and extend it to the case of polynomials with some zeros on an arbitrary analytic curve in the complex plane.

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Abstract  

We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of Al-Salam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.

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