Log-convexity and strong q-log-convexity for some triangular arrays

Zhu, Bao-Xuan

doi:10.1016/j.aam.2012.11.003

Cited by 57 publications

(59 citation statements)

References 28 publications

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“…a k−1 a k+1 ≤ a 2 k ), which is equivalent to that a n−1 a m+1 ≥ a n a m (resp. a n−1 a m+1 ≤ a n a m ) for all 1 ≤ n ≤ m. The log-concave and log-convex sequences arise often in combinatorics, algebra, geometry, analysis, probability and statistics and have been extensively investigated, see Stanley [39], Brenti [8], Liu and Wang [27] and Zhu [45] for details.…”

Section: Introductionmentioning

confidence: 99%

q-log-convexity from linear transformations and polynomials with only real zeros

Zhu

2018

European Journal of Combinatorics

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In this paper, we mainly study the stability of iterated polynomials and linear transformations preserving the strong q-log-convexity of polynomials Let [T n,k ] n,k≥0 be an array of nonnegative numbers. We give some criteria for the linear transformationpreserving the strong q-log-convexity (resp. log-convexity). As applications, we derive that some linear transformations (for instance, the Stirling transformations of two kinds, the Jacobi-Stirling transformations of two kinds, the Legendre-Stirling transformations of two kinds, the central factorial transformations, and so on) preserve the strong q-log-convexity (resp. log-convexity) in a unified manner. In particular, we confirm a conjecture of Lin and Zeng, and extend some results of Chen et al., and Zhu for strong q-log-convexity of polynomials, and some results of Liu and Wang for transformations preserving the log-convexity.The stability property of iterated polynomials implies the q-log-convexity. By applying the method of interlacing of zeros, we also present two criteria for the stability of the iterated Sturm sequences and q-log-convexity of polynomials. As consequences, we get the stabilities of iterated Eulerian polynomials of type A and B, and their q-analogs. In addition, we also prove that the generating functions of alternating runs of type A and B, the longest alternating subsequence and up-down runs of permutations form a q-log-convex sequence, respectively.MSC: 05A20, 11B68

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Section: Introductionmentioning

confidence: 99%

q-log-convexity from linear transformations and polynomials with only real zeros

Zhu

2018

European Journal of Combinatorics

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“…Alternatively, it can be proven by observing that the tridiagonal matrix (2.7) associated to the continued fraction (4.13) is totally positive of order 2, i.e. β n ≥ 0, γ n ≥ 0 and γ n γ n+1 − β n+1 ≥ 0 for all n. This implies [69,82] that (E n+1 ) n≥0 is log-convex. And since E 0 E 2 − E 2 1 = 0, it follows that also (E n ) n≥0 is log-convex.…”

Section: Euler Numbers Partmentioning

confidence: 99%

The Euler and Springer numbers as moment sequences

Sokal

2020

Expositiones Mathematicae

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“…During their long history, they arised often in combinatorics and were extensively studied (see [5,6,7,13] and references therein). In recent years, there has been a considerable amount of interesting extensions and modifications devoted to these polynomials (see [2,3,11,18,19,20,22,25] for instance). In fact, Brenti showed that it is enough to study the Eulerian polynomials for irreducible Coxeter groups [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…As we know that many famous polynomials sequences, such as the Bell polynomials [10,19], the classical Eulerian polynomials [19,25], the Narayana polynomials [9], the Narayana polynomials of type B [8] and the Jacobi-Stirling numbers [17,26], are q-log-convex. Furthermore, almost all of these polynomials sequences are strongly q-logconvex [10,17,25]. In this paper we give the strong q-log-convexity of many Eulerian polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Strong q-log-convexity of the Eulerian polynomials of Coxeter groups

Liu

Zhu

2015

Discrete Mathematics

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a b s t r a c tIn this paper we prove the strong q-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arrays and a criterion for determining the strong q-log-convexity of polynomial sequences, whose generating functions can be given by a continued fraction. As applications, we get the strong q-log-convexity of the Eulerian polynomials of types A n , B n , their q-analogue and the generalized Eulerian polynomials associated to the arithmetic progression {a, a + d, a + 2d, a + 3d, . . .} in a unified manner.

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Log-convexity and strong q-log-convexity for some triangular arrays

Cited by 57 publications

References 28 publications

q-log-convexity from linear transformations and polynomials with only real zeros

q-log-convexity from linear transformations and polynomials with only real zeros

The Euler and Springer numbers as moment sequences

Strong q-log-convexity of the Eulerian polynomials of Coxeter groups

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