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Higher Order Turán Inequalities

Author: Dimitar K. Dimitrov
Journal: Proc. Amer. Math. Soc. 126 (1998), 2033-2037
MSC (1991): Primary 30D10, 33C45
MathSciNet review: 1459117
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Abstract: The celebrated Turán inequalities $P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0, \ \ x \in [-1,1],\ \ n \geq 1$, where $P_{n}(x)$ denotes the Legendre polynomial of degree $n$, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities $\gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0,\ \ n \geq 1$, which hold for the Maclaurin coefficients of the real entire function $\psi$ in the Laguerre-Pólya class, $\psi(x) = \sum _{n=0}^{\infty} \gamma _{n} x^{n}/n!$.

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Additional Information

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

Keywords: Tur\'{a}n inequalities, Tur\'{a}n determinants, entire functions in the Laguerre-P\'{o}lya class, Riemann hypothesis
Received by editor(s): December 12, 1996
Additional Notes: Research supported by the Brazilian foundation CNPq under Grant 300645/95-3 and the Bulgarian Science Foundation under Grant MM-414.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society