On the combinatorics of the Boros-Moll polynomials

Abstract

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.

2-Log-Concavity of the Boros–Moll Polynomials

“…which implies that P m (a) is a polynomial in a with positive coefficients. Chen, Pang and Qu [10] gave a combinatorial argument to show that the double sum (1.2) can be reduced to the single sum (1.3). Let d i (m) be the coefficient of a i of P m (a), that is, Many proofs of the above formula can be found in the survey of Amdeberhan and Moll [2].…”

The Concavity and Convexity of the Boros-Moll Sequences

“…which indicates that the coefficients of a i in P m (a) are positive for 0 i m. Chen, Pang and Qu [12] gave a combinatorial proof to show that (2) is equal to (3). Let d i (m) be defined by…”

“…In fact, the recurrences (11) and (12) are also derived independently by Moll [19] by using the WZ-method [20]. Chen and Gu [11] showed that the Boros-Moll sequences satisfy the reverse ultra log-concavity.…”

Partially 2-colored permutations and the Boros–Moll polynomials

“…Let D i (m) denote the set of all partially 2-colored permutations (π|σ) on [2m] such that the 2-colored permutation π has m + i black elements. For example, consider the partially 2-colored permutation (2,12,8,11,5,9,7,1,4, 3|(6, 10)) in D 2 (6). Then we have A = {1, 8}, B = {2, 3, 4, 5, 7, 9, 11, 12}, and C = {6, 10}.…”

“…The 2-adic valuation of the numbers i!m!2 m+i d i (m) has been studied by Amdeberhan, Manna and Moll [1], and Sun and Moll [16]. By using reluctant functions and an extension of Foata's bijection, Chen, Pang and Qu [9] have found a combinatorial derivation of the single sum formula (1.3) from the double sum formula (1.2). For the special case a = 1, we are led to a combinatorial argument for the identity…”