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Zero distributions via orthogonality

Laurent Baratchart, Reinhold Küstner, Vilmos Totik (2005)

Annales de l’institut Fourier

We develop a new method to prove asymptotic zero distribution for different kinds of orthogonal polynomials. The method directly uses the orthogonality relations. We illustrate the procedure in four cases: classical orthogonality, non-Hermitian orthogonality, orthogonality in rational approximation of Markov functions and its non- Hermitian variant.

Zeros and poles of Dirichlet series

Enrico Bombieri, Alberto Perelli (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Under certain mild analytic assumptions one obtains a lower bound, essentially of order r , for the number of zeros and poles of a Dirichlet series in a disk of radius r . A more precise result is also obtained under more restrictive assumptions but still applying to a large class of Dirichlet series.

Zeros of a certain class of Gauss hypergeometric polynomials

Addisalem Abathun, Rikard Bøgvad (2018)

Czechoslovak Mathematical Journal

We prove that as n , the zeros of the polynomial 2 F 1 - n , α n + 1 α n + 2 ; z cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999–2001) to the case of a complex parameter α and partially proves a conjecture made by the authors in an earlier work.

Zeros of eigenfunctions of some anharmonic oscillators

Alexandre Eremenko, Andrei Gabrielov, Boris Shapiro (2008)

Annales de l’institut Fourier

We study complex zeros of eigenfunctions of second order linear differential operators with real even polynomial potentials. For potentials of degree 4, we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. For potentials of degree 6, we classify eigenfunctions with finitely many zeros, and show that in this case too, all zeros are real or pure imaginary.

Zeros of Sequences of Partial Sums and Overconvergence

Kovacheva, Ralitza K. (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 30B40, 30B10, 30C15, 31A15.We are concerned with overconvergent power series. The main idea is to relate the distribution of the zeros of subsequences of partial sums and the phenomenon of overconvergence. Sufficient conditions for a power series to be overconvergent in terms of the distribution of the zeros of a subsequence are provided, and results of Jentzsch-Szegö type about the asymptotic distribution of the zeros of overconvergent subsequences are stated....

Zeros of solutions of certain higher order linear differential equations

Hong-Yan Xu, Cai-Feng Yi (2010)

Annales Polonici Mathematici

We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation f ( k ) + a k - 1 ( z ) f ( k - 1 ) + + a ( z ) f ' + D ( z ) f = 0 , (1) where D ( z ) = Q ( z ) e P ( z ) + Q ( z ) e P ( z ) + Q ( z ) e P ( z ) , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), a j ( z ) (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

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