Turán-type inequalities and the distribution of zeros of entire functions

Csordás, George

doi:10.1515/9783110282429.25

Cited by 4 publications

(3 citation statements)

References 40 publications

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“…Proofs include transcendental functions and Jensen polynomials. This initiated research in several directions [KS60], including higher Turán inequalities capturing the conjectured hyperbolic property of Jensen polynomials associated with the Riemann zeta function [Cs14,CJW19,GORZ19]. We also refer to the open session report by van Assche [VA10], where Nevai emphasized also the importance of studying Turán inequalities.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Turán inequalities from Chebyshev to Laguerre polynomials

Heim¹,

Neuhäuser²,

Troeger³

2022

Preprint

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Let g and h be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomialswith initial value P g,h 0 (x) = 1 encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov-Okounkov polynomials. In this paper we prove that for g = id and fixed h we obtain orthogonal polynomial sequences for positive definite functionals. Let h(n) = n s with 0 ≤ s ≤ 1. Then the sequence satisfies Turán inequalities for x ≥ 0.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Turán inequalities from Chebyshev to Laguerre polynomials

Heim¹,

Neuhäuser²,

Troeger³

2022

Preprint

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“…( 90), that lets consider an expansion on the Bernoulli numbers. Of course, other infinite sets of relevant numbers like the Gregory coefficients [23] and the Cauchy numbers of the second kind [24] can describe special summations on those ones which converge to the Euler-Mascheroni constant. Now, the evidence shows that three new sets of coefficients a 2n , b n and C n , can contribute to represent this important constant as it has been inferred in the current article.…”

Section: Methodsmentioning

confidence: 99%

New Formulas for the Euler-Mascheroni Constant and other Consequences derived from the Acceptance of Hyperbolicity of Jensen Polynomials and the Analysis of the Turán Moments for the ξ-Function

Mantzakouras,

López

2021

Preprint

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The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Turán moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s = 1/2 + i.t and 3) even coefficients of the Riemann Xi function around s = 1/2. These findings support the acceptance of the property of hyperbolicity of Jensen polynomials within the scope of the Riemann Hypothesis due to exactness on the approximations calculated not only for the Euler-Mascheroni constant, but also for the Bernoulli numbers and the even derivatives of the Riemann Xi function at s = 1/2. The new formulas are linked to similar patterns observed in the formulation of the Akiyama-Tanigawa algorithm based on the Gregory coefficients of second order and lead to understanding the Riemann zeta function as a bridge between the Gregory coefficients and other relevant sets

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“…[9]- [10]. A survey of recent results on Turán type inequalities [7] is published in the proceedings of the conference [8] dedicated to Paul Turán's achievements in different areas of mathematics and applications.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities

Mehrez,

Sitnik

2014

Preprint

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Turán-type inequalities and the distribution of zeros of entire functions

Abstract: The purpose of this note is to survey a select group of classical and new results which highlight the role that the Turán-type inequalities play in the study of the distribution of zeros of polynomials and transcendental entire functions.

Cited by 4 publications

References 40 publications

Turán inequalities from Chebyshev to Laguerre polynomials

Turán inequalities from Chebyshev to Laguerre polynomials

New Formulas for the Euler-Mascheroni Constant and other Consequences derived from the Acceptance of Hyperbolicity of Jensen Polynomials and the Analysis of the Turán Moments for the ξ-Function

Monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities

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