Numerator polynomials of Riordan matrices and generalized Lagrange series.

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. The n th descending diagonal of the ordinary Riordan matrix and the n th descending diagonal of the exponential Riordan matrix have the generating functions respectively g n (φ x) / (1 − φ x) n + 1 and h n (φ x) / (1 − φ x) 2 n + 1 , where g n (x) , h n (x) are polynomials of degree ≤ n. We will call these polynomials the numerator polynomials of Riordan matrices. General properties of these polynomials were considered in separate paper. In this paper, we will consider numerator polynomials of the Riordan matrices associated with the family of series a (β) (x) = a (x (β) a β (x)). The matrices of transformations, in which these polynomials participate, have the form A n E n β A n − 1 , where A n is the certain matrix of order n + 1 , E is the matrix of the shift operator. The main focus is on studying the properties of these matrices. [ABSTRACT FROM AUTHOR]