Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let $P(x)$ be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of $P(x+1)$, which leads to the log-concavity of $P(x+c)$ for any $c\geq 1$ due to Llamas and Martinez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.

[1]  Victor H. Moll,et al.  An Extension of a Criterion for Unimodality , 2001, Electron. J. Comb..

[2]  Miguel Angel Fiol,et al.  On the algebraic theory of pseudo-distance-regularity around a set☆ , 1999 .

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  A. Hoffman.  On the Polynomial of a Graph , 1963 .

[5]  Victor H. Moll.  The Evaluation of Integrals: A Personal Story , 2002 .

[6]  Paul Terwilliger.  An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph , 2004 .

[7]  Manuel Kauers,et al.  A Computer Proof of Moll’s Log-Concavity Conjecture , 2007 .

[8]  José Martínez-Bernal,et al.  Nested Log-Concavity , 2010 .

[9]  R. Stanley.  Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a , 1989 .

[10]  William Y. C. Chen,et al.  The ratio monotonicity of the Boros-Moll polynomials , 2008, Math. Comput..

[11]  Paul M. Terwilliger.  The Subconstituent Algebra of an Association Scheme, (Part I) , 1992 .

[12]  Yeong-Nan Yeh,et al.  Proof of a conjecture on unimodality , 2005, Eur. J. Comb..

[13]  Victor H. Moll,et al.  A Criterion for Unimodality , 1999, Electron. J. Comb..

[14]  Hiroshi Suzuki,et al.  The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph , 2005 .

[15]  V. Moll,et al.  A Sequence of Unimodal Polynomials , 1999 .

[16]  Miguel Angel Fiol,et al.  Locally Pseudo-Distance-Regular Graphs , 1996, J. Comb. Theory, Ser. B.

[17]  F. Brenti,et al.  Unimodal, log-concave and Pólya frequency sequences in combinatorics , 1989 .

[18]  Miguel Angel Fiol.  On pseudo-distance-regularity , 2001 .

[19]  William Y. C. Chen,et al.  The ratio monotonicity of the q-derangement numbers , 2011, Discret. Math..

[20]  N. Biggs.  Algebraic Graph Theory: COLOURING PROBLEMS , 1974 .

[21]  Miguel Angel Fiol,et al.  An Algebraic Characterization of Completely Regular Codes in Distance-Regular Graphs , 2001, SIAM J. Discret. Math..

[22]  Miguel Angel Fiol,et al.  Some Families of Orthogonal Polynomials of a Discrete Variable and their Applications to Graphs and Codes , 2009, Electron. J. Comb..