Determinant Representations of Polynomial Sequences of Riordan Type

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given. 1. Introduction The concept of a Riordan array is very useful in combinatorics. The infinite triangles of Pascal, Catalan, Motzkin, and Schr？der are important and meaningful examples of Riordan array, and many others have been proposed and developed (see, e.g., [1–7]). In the recent literature, Riordan arrays have attracted the attention of various authors from many points of view and many examples and generalizations can be found (see, e.g., [8–12]). A Riordan array denoted by is an infinite lower triangular matrix such that its column ？？( ) has generating function , where and are formal power series with , , and . That is, the general term of matrix is ; here denotes the coefficient of in power series . Given a Riordan array and column vector , the product of and gives a column vector whose generating function is , where . If we identify a vector with its ordinary generating function, the composition rule can be rewritten as This property is called the fundamental theorem for Riordan arrays and this leads to the matrix multiplication for Riordan arrays: The set of all Riordan arrays forms a group under the previos operation of a matrix multiplication. The identity element of the group is . The inverse element of is where is compositional inverse of . A Riordan array can be characterized by two sequences and such that, for If and are the generating functions for the - and -sequences, respectively, then it follows that [9, 13] If the inverse of is , then the - and -sequences of are For an invertible lower triangular matrix , its production matrix (also called its Stieltjes matrix; see [11, 14]) is the matrix , where is the matrix with its first row removed. The production matrix can be characterized by the matrix equality , where ( is the usual Kronecker delta). Lemma 1 (see [14]). Assume that is an infinite lower triangular matrix with . Then is a Riordan array if and only if its production matrix is of the form where is the A-sequence and is the Z-sequence of the Riordan array . Definition 2. Let be a sequence of polynomials where is of degree and . We say that is a polynomial sequence of Riordan type if the coefficient matrix is an element