Q-log-concavity and log-convexity for q-analogue of polynomial coefficient

The aim of this paper is twofold. First, we propose a q-analogue of polynomial coeffcients ((n)(k))(a), associated with the vector a = (a(0), ..., a(s)), which are defined as the k-th coefficients of (a(0) + a(1)x + ... + a(s)x(s))(n). Second, we prove that these coefficients are log-concave and their triangle is total positive of order two matrix (classical and analog versions). We also show that their transformation Sigma(sn)(k=0)((n)(k))(a) X-k (resp. Sigma(sn)(k=0)[(n)(k)](a) X-k) preserve the log-convexity property when the sequence (a(0), ..., a(s)) is log-concave.