We establish the preserving log-convexity property for the generalized Pascal triangles. It is an extension of a result of H. Davenport and G. Pólya “On the product of two power series”, who proved that the binomial convolution of two log-convex sequences is log-convex.

We investigate coordination numbers of the cubic lattices with emphases on their analytic behaviors, including total positivity matrices, distribution zeros polynomials, asymptotic normality coefficients log-concavity and log-convexity numbers.

In this paper, we give a sufficient condition for the linear transformation preserving the strong q-log-convexity. As applications, we get some linear transformations (for instance, Morgan-Voyce transformation, binomial transformation, Narayana transformations of two kinds) preserving the strong q-log-convexity. In addition, our results not only extend some known results, but also imply the str...

Abstract In this paper, we prove that the sequences of Stirling numbers first and second kind with higher level are both Pólya frequency log-concave. Then, show some polynomials related to above q -log-convex or strongly -log-convex. Furthermore, establish linear transformation 2 preserves log-convexity.

An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...

Borwein, Affleck, and Girgensohn (2000) posed a problem concerning the shape (that is, convexity, log-convexity, reciprocal concavity) of a certain function of several arguments that had manifested in a number of contexts concerned with optimization problems. In this paper we further explore the shape of the Borwein-Affleck-Girgensohn function and especially its extensions generated by complete...