Recurrence Relations for Strongly <i>q</i>-Log-Convex Polynomials

Chen, William Y. C.; Wang, Larry X. W.; Yang, Arthur L. B.

doi:10.4153/cmb-2011-008-5

Cited by 43 publications

(40 citation statements)

References 26 publications

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“…The q-log-convexity and strong q-log-convexity of the row generating functions S n (q) = n k=0 S n,k q k have been proved, see Liu and Wang [27], Chen et al [14] and Zhu [45,46] for instance. Liu and Wang [27] also proved that both the Stirling transformations of two kinds preserve the log-convexity.…”

Section: Stirling Transformations Of Two Kindsmentioning

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: Stirling Transformations Of Two Kindsmentioning

confidence: 99%

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

Zhu

2018

European Journal of Combinatorics

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“…has nonnegative coefficients as a polynomial in q. It has been shown that many combinatorial polynomials are q-log-convex, such as the Bell polynomials, the Eulerian polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials, see Liu and Wang [17], and Chen, Wang and Yang [7]. Meanwhile, it should be noticed that Butler and Flanigan [5] gave a different notion of q-log-convexity in their study of q-Catalan numbers, namely, a sequence of polynomials (f k (q)) k≥0 is called q-log-convex if f m−1 (q)f n+1 (q) − q n−m+1 f m (q)f n (q) has nonnegative coefficients for n ≥ m ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

The q-log-convexity of the Narayana polynomials of type B

Chen

Tang

Wang

et al. 2010

Advances in Applied Mathematics

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We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. By the principal specialization this, leads to q-log-convexity. We also show that the linear transformation with respect to the triangular array of Narayana numbers of type B is log-convexity preserving.

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“…It is known that many polynomials, denoted by A n (q) = n k=0 A n,k q k for n 0, have strong q-log-convexity (see [18,42]). Note that the sequence {q k } k 0 is strongly q-log-convex.…”

Section: Strong Q-log-convexity and Linear Transformationsmentioning

confidence: 99%

“…Corollary 4.8 (Chen et al . [18]). The polynomials D n (q), F n,m (q) and G n,m (q) are all strongly q-log-convex.…”

Section: Strong Q-log-convexity and Linear Transformationsmentioning

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

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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.

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Recurrence Relations for Strongly q-Log-Convex Polynomials

Cited by 43 publications

References 26 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 q-log-convexity of the Narayana polynomials of type B

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

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