HIGHER ORDER TURAN INEQUALITIES FOR THE RIEMANN xi-FUNCTION

Author(s)

Dimitrov, Dimitar K. ^[1] ; Lucas, Fabio R. ^[2]

Affiliation

^[1] Univ Estadual Paulista, IBILCE, Dept Ciencias Computacao & Estat, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

^[2] Univ Estadual Campinas, IMECC, Dept Matemat Aplicada, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Proceedings of the American Mathematical Society; v. 139, n. 3, p. 1013-1022, MAR 2011.
• Web of Science Citations: 7

Abstract

The simplest necessary conditions for an entire function psi(x) = Sigma(infinity)(k=0) gamma k xk/k! to be in the Laguerre-Polya class are the Turan inequalities gamma(2)(k) - gamma k+1 gamma k-1 >= 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with psi to be hyperbolic. The higher order Turan inequalities 4(gamma(2)(n) - gamma n-1 gamma n+1)(gamma(2)(n+1), - gamma n gamma n+2) - (gamma n gamma n+1 - gamma n-1 gamma n+2)(2) >= 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Polya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Polya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turan inequalities hold for the coefficients of the Riemann xi-function. In this short paper, we prove that the higher order Turan inequalities also hold for the xi-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. (AU)

• FAPESP's process: 03/01874-2 - Orthogonal and similar polynomials: properties and applications
• Grantee: Alagacone Sri Ranga
• Support Opportunities: Research Projects - Thematic Grants