Some positivities in certain triangular arrays

Zhu, Bao-Xuan

doi:10.1090/s0002-9939-2014-12008-9

Cited by 33 publications

(29 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Theorem 1.3, we have the following generalization, which in particular confirms Conjecture 1.1 for Jacobi-Stirling numbers of the second kind. [46], we derive that [Jc k n (z)] n,k is totally positive, also see [33]. Thus by Theorem 1.2, we also have the following result, which in particular confirms Conjecture 1.1 for Jacobi-Stirling numbers of the first kind.…”

Section: Jacobi-stirling Transformation Of the Second Kindsupporting

confidence: 77%

See 1 more Smart Citation

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

Zhu

2018

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Jacobi-stirling Transformation Of the Second Kindsupporting

confidence: 77%

“…Zhu [46] proved that the row generating functions of U(n, k) (respectively, V (n, k)) form a strongly q-log-convex sequence. In view of Theorem 1.3, these can be extended to the following result.…”

Section: Central Factorial Transformationsmentioning

confidence: 99%

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

Zhu

2018

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Bell polynomials are given by B n+1 (q) = k 0 S n+1,k x k , where S n,k are the Stirling numbers of the second kind. It has been shown that the polynomials B n (q) form a strongly q-log-convex sequence (see, for instance, [18,42,43]). Note that B n+1 (q) = n k=0 n k B n (q).…”

Section: B-x Zhumentioning

confidence: 99%

“…where T n,k = 0 unless 0 k n. Two known results for the strong q-log-convexity of the generating functions of row sequences of [T n,k ] n,k 0 , for a 1 = b 1 = 0 and for A = B = 0, respectively, were proved by Chen et al . [18] and Zhu [43]. Therefore, we also consider a similar problem for the above special interesting Riordan arrays (see theorem 3.4).…”

Section: Introductionmentioning

confidence: 99%

Log-concavity and strong q-log-convexity for Riordan arrays and recursive matrices

Zhu¹

2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

View full text Add to dashboard Cite

Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrencefor n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not only preserves the strong q-log-convexity property, but also preserves the strong q-log-concavity property. Finally, we demonstrate that the strong q-log-convexity property is preserved by the Stirling transformation and Whitney transformation of the second kind, which extends some known results for the strong q-log-convexity property.

show abstract

“…However, the converse dose not follows. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Liu

Zhu

2015

Discrete Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Some positivities in certain triangular arrays

Cited by 33 publications

References 22 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

Log-concavity and strong q-log-convexity for Riordan arrays and recursive matrices

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

Contact Info

Product

Resources

About