Row polynomial matrices of Riordan arrays

Let R=[rn,k]n,k≥0 be a Riordan array. Define the row polynomials Rn(q)=∑k=0nrn,kqk and the row polynomial matrix R(q)=[rn,k(q)]n,k≥0 by rn,k(q)=∑j=knrn,jqj−k. Then R(q) is also a Riordan array with the Rn(q) located on the leftmost column of R(q). In this paper we investigate combinatorial properties of the matrix R(q) and the sequence (Rn(q))n≥0, including their characterizations, the q-total positivity of R(q) and the q-log-convexity of (Rn(q))n≥0.