HIGHER ORDER TURAN INEQUALITIES FOR THE RIEMANN xi-FUNCTION - Dimitrov, Dimitar Kolev [UNESP]; Lucas, Fabio R. (Amer Mathematical Soc, 2011)
The simplest necessary conditions for an entire functionpsi(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.
Keywords: Laguerre-Polya class; Maclaurin coefficients; Turan inequalities; Jensen polynomials; Riemann xi function