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

William Y.C. Chen, Robert L. Tang, Larry X.W. Wang, Arthur L.B. Yang
2010 Advances in Applied Mathematics  
We prove a conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials of type B.  ...  We also show that the linear transformation with respect to the triangular array of Narayana numbers of type B is log-convexity preserving. ( where d is the rank of A n and h(q) is a polynomial of degree  ...  This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, and the National Science Foundation of China.  ... 
doi:10.1016/j.aam.2009.03.004 fatcat:n4bqc4bhdrhajncrhzcmycjjw4

Schur positivity and the q-log-convexity of the Narayana polynomials

William Y. C. Chen, Larry X. W. Wang, Arthur L. B. Yang
2010 Journal of Algebraic Combinatorics  
We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity.  ...  Moreover, we prove that the Narayana polynomials are strongly q-log-convex.  ...  This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, and the National Science Foundation of China.  ... 
doi:10.1007/s10801-010-0216-x fatcat:zk3yzajrtrdalkbcwv66rtnxau

Schur Positivity and the q-Log-convexity of the Narayana Polynomials [article]

William Y. C. Chen, Larry X.W. Wang, Arthur L. B. Yang
2008 arXiv   pre-print
the product of Schur functions, and obtain the strong q-log-convexity of the Narayana polynomials and the strong q-log-concavity of the q-Narayana numbers.  ...  Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the q-log-convexity of the Narayana polynomials, and a proof of  ...  This work was supported by the 973 Project, the PC-SIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.  ... 
arXiv:0806.1561v1 fatcat:5wggi267wfbhfl2b2zy7mtnomy

Positivity of iterated sequences of polynomials [article]

Bao-Xuan Zhu
2018 arXiv   pre-print
, 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  ...  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  ...  In addition, results of this paper were presented in the Institute of Mathematics, Academia Sinica, Taipei in Jan. 2016, the seventh National Conference on Combinatorics and Graph Theory at Hebei Normal  ... 
arXiv:1807.01062v1 fatcat:qv2tlw5m4zeh5jkrd24xf3dsti

Log-convex and Stieltjes moment sequences [article]

Yi Wang, Bao-Xuan Zhu
2016 arXiv   pre-print
We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences.  ...  the infinite log-convexity of the Schröder numbers.  ...  Acknowledgements This work was supported in part by the National Natural Science Foundation of China (Nos. 11201191, 11371078, 11571150) .  ... 
arXiv:1612.04114v1 fatcat:4kdv75hd6jhevo5grledspbdtq

Strong q-log-convexity of the Eulerian polynomials of Coxeter groups [article]

Lily Li Liu, Bao-Xuan Zhu
2014 arXiv   pre-print
As consequences, we get the strong q-log-convexity the Eulerian polynomials of type A_n,B_n, their q-analogous and the generalized Eulerian polynomials associated to the arithmetic progression {a,a+d,a  ...  In this paper we prove the strong q-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions.  ...  The polynomials P (B n , q) form a strongly q-log-convex sequence. From the definitions, if a sequence of polynomials is strongly q-log-convex, then it is q-log-convex.  ... 
arXiv:1407.1968v2 fatcat:dfe5uid64ffolgpoyxuufp77ru

Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Log-concavity [article]

A.N. Kirillov
2015 arXiv   pre-print
In fact we prove a combinatorial formula for generalized q-Gaussian polynomials which is a far generalization of the so-called KOH-identity O, as well as it manifests the unimodality property of the q-Gaussian  ...  Another application of the Rigged Configuration technique presented, is a new family of counterexamples to Okounkov's log-concavity conjecture Ok.  ...  Narayana numbers) N(n, m; k | q) = k a=0 (−1) k−a q ( k−a 2 ) n m + 1 k − a q n−1 b=0 [b]!  ... 
arXiv:1505.01542v1 fatcat:l2pprtdeyjdclgwhzdjfribvzq

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

Bao-Xuan Zhu
2018 European journal of combinatorics (Print)  
As consequences, we get the stabilities of iterated Eulerian polynomials of type A and B, and their q-analogs.  ...  The stability property of iterated polynomials implies the q-log-convexity.  ...  triangle of the second kind, the Jacobi-Stirling triangle of the second kind, the Legendre-Stirling triangle of the second kind, the Eulerian triangles of Types A and B, the Narayana triangles of Types  ... 
doi:10.1016/j.ejc.2018.06.003 fatcat:xe6fwgy5pbglvcqhoegal56hue

The q-log-convexity of Domb's polynomials [article]

Donna Q.J. Dou, Anne X.Y. Ren
2013 arXiv   pre-print
Our proof is based on the q-log-convexity of Narayana polynomials of type B and a criterion for determining q-log-convexity of self-reciprocal polynomials.  ...  In this paper, we prove the q-log-convexity of Domb's polynomials, which was conjectured by Sun in the study of Ramanujan-Sato type series for powers of π.  ...  This work was supported by the 973 Project, the PC-SIRT Project of the Ministry of Education, the National Science Foundation of China, and the Fundamental Research Funds for the Central Universities of  ... 
arXiv:1308.2961v1 fatcat:kki6bpgn5ndyjf7exsrifjjxv4

On a Stirling-Whitney-Riordan triangle [article]

Bao-Xuan Zhu
2021 arXiv   pre-print
We show that the row-generating function T_n(q) has only real zeros and the Turán-type polynomial T_n+1(q)T_n-1(q)-T^2_n(q) is stable.  ...  Furthermore, we get the x-Stieltjes moment property and 3-x-log-convexity of T_n(q) and show that the triangular convolution z_n=∑_i=0^nT_n,ix_iy_n-i preserves Stieltjes moment property of sequences.  ...  , the Narayana polynomials of type A and B, Jacobi-Stirling polynomials, and so on (see Liu and Wang [26] , Chen et al  ... 
arXiv:2008.04120v4 fatcat:nzcoytawszcm7oferq2m2aznyq

Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle [article]

Bao-Xuan Zhu
2022 arXiv   pre-print
positive and log-convex of higher order in all the indeterminates.  ...  Using the theory of production matrices, we gain a criterion for the coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the m-Jacobi-Rogers triangle.  ...  For example, the Bell polynomials, the classical Eulerian polynomials, the Narayana polynomials of type A and B, Ramanujan polynomials, Dowling polynomials, Jacobi-Stirling polynomials, and so on, are  ... 
arXiv:2202.03793v1 fatcat:vlcl67ceyrd4jpswirdc23qo6m

A unified approach to combinatorial triangles: a generalized Eulerian polynomial [article]

Bao-Xuan Zhu
2020 arXiv   pre-print
Finally, we apply our results to an array from the Lambert function, a triangular array from staircase tableaux and the alternating-runs triangle of type B in a unified approach.  ...  Then we extend the famous Frobenius formula, the γ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial.  ...  Many famous polynomials were proved to be q-log-convex or strongly q-log-convex, e.g., the Bell polynomials, the classical Eulerian polynomials, the Narayana polynomials of type A and B, Jacobi-Stirling  ... 
arXiv:2007.12602v1 fatcat:kbpwvccufrgdjnlanh7c5ujpme

Calculus proofs of some combinatorial inequalities [article]

Tomislav Došlić, Darko Veljan
2006 arXiv   pre-print
Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity.  ...  Our method also applies to show that sequences of values of some orthogonal polynomials, and in particular the sequence of central Delannoy numbers, are log-convex.  ...  The sequence P n (t) of the values of Legendre polynomials is log-convex for any fixed real t ≥ 1.  ... 
arXiv:math/0603405v1 fatcat:3i2qetxlifbnnds5kx6c2ulw6q

Total positivity from the exponential Riordan arrays [article]

Bao-Xuan Zhu
2021 arXiv   pre-print
Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser.  ...  A 85 (1999)] extended them to q-log-concavity and almost q-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix.  ...  ] , Narayana triangles of types A and B [61] , and the generalized Jacobi-Stirling triangle [67] .  ... 
arXiv:2006.14485v3 fatcat:zs4jcxqtujgb3g5gml5syvhnku

The q-Log-convexity of the Generating Functions of the Squares of Binomial Coefficients [article]

William Y. C. Chen , Larry X. W. Wang
2008 arXiv   pre-print
We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence {∑_k=0^nn k^2q^k}_n≥ 0.  ...  Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients.  ...  This work was supported by the 973 Project, the PC-SIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.  ... 
arXiv:0810.2247v1 fatcat:l3zi2oijwvb57a3b2p6uxr3lle
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