The Unimodal Nature Of The Polynomial Sequence And The Q-Stieltjes Moment Property

Unimodality problem is one of the main topics in combinatorics, including the study of log-convexity (log-concavity), q-log-concavity, Totally Positive (TP for short) and unimodality. In this paper we will study the unimodality of the sequence and the q-Stieltjes moment (SM for short) property. The study of the q-log-concavity and log-convexity (log-concavity) is main content in unimodality problem, and with the rapid development of combinatorial mathematics, this problem has become more and more important in mathematics. So it is worth studying. And the q-Stieltjes moment property is one of the important property in combinatorial sequence. And it connects the functions to the sequences. Therefore, it has research value.In this paper, we mainly study two parts.The first part we consider unimodality problems of the sequence of q-binomial co-efficients located in a ray or a transversal of the q-Pascal triangle. We present the log-convexity of the sequence of q-binomial coefficients located in a ray or a transversal of the q-Pascal triangle using Su and Wang's results for binomials coefficients. Then we give the q-log-concavity of the the sequence of q-binomial coefficients located in a ray or a transversal of the q-Pascal triangle.The second part, we discuss the q-Stieltjes moment property of some polynomials related to Dowling lattices using their exponential generating functions. We first give the continued fraction expression of the general polynomial sequences based on the theory of exponential Riordan array and orthogonal polynomial. As applications, we can obtain the continued fraction of the Bell polynomials, the Dowling polynomials, the r-Bell and the r-Dowling polynomials. Then we obtain the q-Stieltjes moment property of these polynomials in a unified manner from the continued fraction expression and a criterion for determining the q-Stieltjes moment property of polynomial sequences. Finally, we present some convolutions that preserve the Stieltjes moment property.