The n-th production matrix of a Riordan array

The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the n-th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the n-th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every n-th production matrix can be factorized into a product of n matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the n-th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the n-th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases.