Boros-Moll Polynomials Research Articles

The real-rootedness of generalized Narayana polynomials related to the Boros-Moll polynomials

In this paper, we prove the real-rootedness of a family of generalized Narayana polynomials which arose in the study of the infinite log-concavity of the Boros-Moll polynomials. We establish certain recurrence relations for these Narayana polynomials, from which we derive the real-rootedness. In order to prove the real-rootedness, we use a sufficient condition due to Liu and Wang to determine whether two polynomials have interlaced zeros. The recurrence relations are verified with the help of the $Mathematica$ package $HolonomicFunctions$.

Interlacing log-concavity of the Boros–Moll polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence {Pm(x)}m≥0, where Pm(x) is a polynomial of degree m with positive coefficients. This sequence is said to be interlacingly log-concave if the ratios of consecutive coefficients of Pm(x) interlace the ratios of consecutive coefficients of Pm+1(x) for any m ≥ 0. The interlacing log-concavity of a sequence of polynomials is stronger than the log-concavity of the polynomials themselves. We show that the Boros-Moll polynomials are interlacingly log-concave. Furthermore, we give a sufficient condition for the interlacing log-concavity which implies that some classical combinatorial polynomials are interlacingly log-concave.

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 Martínez-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.

On the combinatorics of the Boros-Moll polynomials

The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan’s Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata’s bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.

The reverse ultra log-concavity of the Boros-Moll polynomials

We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence { i ! d i ( m ) } \{i!d_i(m)\} for any m ≥ 2 m\geq 2 , where d i ( m ) d_i(m) are the coefficients of the Boros-Moll polynomials P m ( a ) P_m(a) . This inequality also leads to the fact that in the asymptotic sense, the Boros-Moll sequences are just on the borderline between ultra log-concavity and reverse ultra log-concavity. We propose two conjectures on the log-concavity and reverse ultra log-concavity of the sequence { d i − 1 ( m ) d i + 1 ( m ) / d i ( m ) 2 } \{d_{i-1}(m) d_{i+1}(m)/d_i(m)^2\} for m ≥ 2 m\geq 2 .

The ratio monotonicity of the Boros-Moll polynomials

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 ∞ \infty -log-concavity.