Total positivity of Riordan arrays

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 and the log-concavity of each row. We present sufficient conditions for the total positivity of Riordan arrays. As applications, two classes of special Riordan arrays are considered. One is related to the recursive matrices introduced by M. Aigner [1], the other is the consistent Riordan arrays introduced in [2]. We show that many well-known combinatorial triangles are totally positive and many famous combinatorial numbers are log-convex in a unified approach. In addition, we give a combinatorial interpretation of the log-convexity of the 0th column of a Riordan array.