| The combinatorial sequences have many important properties, such as log-convexity,log-concavity, P(?)olya frequence(PF for short) property, and Stieltjes moment property.In this paper we study the strong q-log-convexity of the polynomials sequences and the infinite log-monotonicity of the sequences. The strong q-log-convexity is obtained based on the log-convexity, then a series of important conclusions have been obtained since many scholars studied it. So it is worth studying. While the infinite log-monotonicity of sequences is proposed by the completely monotonic functions, and the notion connects the completely monotonic functions to the infinitely log-monotonic sequences. Therefore,it has research value.In this paper we mainly study two parts.The first part discusses the strong q-log-convexity of the polynomials sequences. First,we present the strong q-log-convexity of the Eulerian-Dowling polynomials using their exponential generating functions. Our proof is based on the theory of exponential Riordan arrays and a criterion for determining the strong q-log-convexity of polynomials sequences,whose generating functions can be given by a continued fraction. Then we give the sufficient conditions for the linear transformation preserving the strong q-log-convexity of polynomials sequences. As applications, we obtain that the r-Dowling polynomials sequences are strongly q-log-convexity.The second part discusses the infinite log-monotonicity of the sequences. At first, we show that the infinite log-monotonicity of the sequences is preserved under the componentwise product. As applications, we can obtain that the central column sequences of some classical combinatorial triangles, such as the Catalan triangles and the Narayana triangle,are infinitely log-monotonic respectively. At last, we give the notion of infinite reverse logmonotonicity of the sequences. And then we present the infinite reverse log-monotonicity of the sequences located along a ray in the q-Pascal triangle. |
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