Brändén's Conjectures on the Boros–Moll Polynomials

Chen, William Y. C.; Dou, Donna Q. J.; Yang, Arthur L. B.

doi:10.1093/imrn/rns193

Cited by 13 publications

(7 citation statements)

References 16 publications

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“…Moll [32] conjectured that the coefficients of P n (x) form a log-concave sequence, which was proved by Kauers and Paule [27]. Moreover, as an extension, Chen et.al [20,16] proved that 2-log-concavity and 3-log-concavity of the coefficients of P n (x) and confirmed two Brändén's conjectures on real roots of polynomials related to P n (x) [8]. However, the stronger conjecture on the infinitely log-concavity of the coefficients of P n (x) proposed by Boros and Moll [7] is still open.…”

Section: Motivationsmentioning

confidence: 99%

Positivity of Iterated Sequences of Polynomials

Zhu¹

2018

SIAM J. Discrete Math.

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In this paper, we present some criteria for the 2-q-log-convexity and 3-q-logconvexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array [A n,k (q)] n,k≥0 of polynomials in q with nonnegative coefficients satisfying the recurrence relationThose criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the 2-q-log-convexity of alternating Eulerian polynomials, 2-log-convexity of Euler numbers, and 3-q-log-convexity of many classical polynomials, including the Bell polynomials, the Eulerian polynomials of Types A and B, the q-Schröder numbers, q-central Delannoy numbers, the Narayana polynomials of Types A and B, the generating functions of rows in the Catalan triangles of Aigner and Shapiro, the generating functions of rows in the large Schröder triangle, and so on, which extend many known results for q-log-convexity.MSC: 05A20; 05A30; 11B37; 11B83; 30B70

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Section: Motivationsmentioning

confidence: 99%

Positivity of Iterated Sequences of Polynomials

Zhu¹

2018

SIAM J. Discrete Math.

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“…We believe that this log-concavity problem could be of independent interest (see, for example, [2,3,15] for similar problems arising in combinatorics).…”

Section: Lemma 315 We Have Qmentioning

confidence: 99%

On the KNS Conjecture in Type E

Gleitz

2014

Ann. Comb.

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For the exceptional types E 6 , E 7 , and E 8 , we prove that the specializations at roots of unity of the quantum dimensions of the Kirillov-Reshetikhin modules give real solutions of -restricted Q-systems, as conjectured by Kuniba, Nakanishi, and Suzuki [9]. We also show that these solutions are positive in type E 6 . In types E 7 and E 8 , we only prove positivity for a subset of the nodes of the Dynkin diagram.

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“…He conjectured the real-rootedness of some variations of P n (x), from which its 3-fold logconcavity can be deduced. Brändén's conjectures were later confirmed by Chen, Dou and Yang [7]. While Conjecture 1.2 is open, Brändén [5] has proved the infinite log-concavity of real-rooted polynomials, which was independently conjectured by Stanley, McNamara and Sagan [16], and Fisk [9].…”

Section: Introductionmentioning

confidence: 99%

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

Chen¹,

Yang²,

Zhang³

2018

Rocky Mountain J. Math.

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In this paper, we prove the real-rootedness of two classes of generalized Narayana polynomials: one arising as the h-polynomials of the generalized associahedron associated to the finite Weyl groups, the other arising in the study of the infinite log-concavity of the Boros-Moll polynomials. For the former, Brändén has already proved that these hpolynomials have only real zeros. We establish certain recurrence relations for the two classes of Narayana polynomials, from which we derive the real-rootedness. 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.

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Brändén's Conjectures on the Boros–Moll Polynomials

Cited by 13 publications

References 16 publications

Positivity of Iterated Sequences of Polynomials

Positivity of Iterated Sequences of Polynomials

On the KNS Conjecture in Type E

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

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