2016/05/01 by Zhu, Bao-Xuan
#05A10 #05A20 #11B73 #15B36 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1605.00257
It is well-known that the binomial transformation preserves the log-concavity property and log-convexity property. Let \binoma+nb+k be the binomial coefficients and \binomn,kj be defined by (b0+b1x+⋯+bkxk)n:=∑j=0kn\binomn,kjxj, where the sequence (bi)0≤ i≤ k is log-concave. In this paper, we prove that the linear transformation yn(q)=∑k=0n\binoma+nb+kxk(q) preserves the strong q-log-concavity property for any fixed nonnegative integers a and b, which strengthens and gives a simple proof of results of Ehrenborg and Steingrimsson, and Wang, respectively, on linear transformations preserving the log-concavity property. We also show that the linear transformation yn=∑i=0kn\binomn,kjxi not only preserves the log-concavity property, but also preserves the log-convexity property, which extends the results of Ahmia and Belbachir about the s-triangle transformation preserving the log-convexity property and log-concavity property. Let [An,k(q)]n, k≥0 be an infinite lower triangular array of polynomials in q with nonnegative coefficients satisfying the recurrence An,k(q)=fn,k(q) An-1,k-1(q)+gn,k(q) An-1,k(q)+hn,k(q) An-1,k+1(q), for n≥ 1 and k≥ 0, where A0,0(q)=1, A0,k(q)=A0,-1(q)=0 for k>0. We present criterions for the strong q-log-concavity of the sequences in each row of [An,k(q)]n, k≥0. As applications, we get the strong q-log-concavity or the log-concavity of the sequences in each row of many well-known triangular arrays, such as the Bell polynomials triangle, the Eulerian polynomials triangle and the Narayana polynomials triangle in a unified approach.