The Ratio Monotonicity of the Boros-Moll Polynomials

William Y.C. Chen and Ernest X.W. Xia

  Abstract:  In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the -log-concavity.

  AMS Classification:  05A20; 33F10

  Keywords:  ratio monotone property, spiral property, unimodality, log-concavity, Jacobi polynomials, Boros-Moll polynomials

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