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

2010
*
Advances in Applied Mathematics
*

We prove a conjecture

doi:10.1016/j.aam.2009.03.004
fatcat:n4bqc4bhdrhajncrhzcmycjjw4
*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. ...##
###
Schur positivity and the q-log-convexity of the Narayana polynomials

2010
*
Journal of Algebraic Combinatorics
*

We prove two recent conjectures

doi:10.1007/s10801-010-0216-x
fatcat:zk3yzajrtrdalkbcwv66rtnxau
*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. ...##
###
Schur Positivity and the q-Log-convexity of the Narayana Polynomials
[article]

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. ...

##
###
Positivity of iterated sequences of polynomials
[article]

2018
*
arXiv
*
pre-print

,

arXiv:1807.01062v1
fatcat:qv2tlw5m4zeh5jkrd24xf3dsti
*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 ...##
###
Log-convex and Stieltjes moment sequences
[article]

2016
*
arXiv
*
pre-print

We introduce

arXiv:1612.04114v1
fatcat:4kdv75hd6jhevo5grledspbdtq
*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) . ...##
###
Strong q-log-convexity of the Eulerian polynomials of Coxeter groups
[article]

2014
*
arXiv
*
pre-print

As consequences, we get

arXiv:1407.1968v2
fatcat:dfe5uid64ffolgpoyxuufp77ru
*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*. ...##
###
Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Log-concavity
[article]

2015
*
arXiv
*
pre-print

In fact we prove a combinatorial formula for generalized

arXiv:1505.01542v1
fatcat:l2pprtdeyjdclgwhzdjfribvzq
*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*]! ...##
###
q-log-convexity from linear transformations and polynomials with only real zeros

2018
*
European journal of combinatorics (Print)
*

As consequences, we get

doi:10.1016/j.ejc.2018.06.003
fatcat:xe6fwgy5pbglvcqhoegal56hue
*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*...##
###
The q-log-convexity of Domb's polynomials
[article]

2013
*
arXiv
*
pre-print

Our proof is based on

arXiv:1308.2961v1
fatcat:kki6bpgn5ndyjf7exsrifjjxv4
*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*...##
###
On a Stirling-Whitney-Riordan triangle
[article]

2021
*
arXiv
*
pre-print

We show that

arXiv:2008.04120v4
fatcat:nzcoytawszcm7oferq2m2aznyq
*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 ...##
###
Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle
[article]

2022
*
arXiv
*
pre-print

positive and

arXiv:2202.03793v1
fatcat:vlcl67ceyrd4jpswirdc23qo6m
*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 ...##
###
A unified approach to combinatorial triangles: a generalized Eulerian polynomial
[article]

2020
*
arXiv
*
pre-print

Finally, we apply our results to an array from

arXiv:2007.12602v1
fatcat:kbpwvccufrgdjnlanh7c5ujpme
*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 ...##
###
Calculus proofs of some combinatorial inequalities
[article]

2006
*
arXiv
*
pre-print

Using calculus we show how to prove some combinatorial inequalities

arXiv:math/0603405v1
fatcat:3i2qetxlifbnnds5kx6c2ulw6q
*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. ...##
###
Total positivity from the exponential Riordan arrays
[article]

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] . ...

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

2008
*
arXiv
*
pre-print

We prove a conjecture

arXiv:0810.2247v1
fatcat:l3zi2oijwvb57a3b2p6uxr3lle
*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. ...
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