Abstract

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 Schröder are important and meaningful examples of Riordan array, and many others have been proposed and developed (see, e.g., [17]). 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., [812]).

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 of the Riordan group; that is, there exists a Riordan array such that . In this case, we say that is the polynomial sequence associated with the Riordan array .
Letting , , then in matrix form we have Hence, by using (1), we have the following lemma.

Lemma 3. Let be the polynomial sequence associated with a Riordan array , and let be its generating function. Then

In [15], Luzón introduced a new notation to represent the Riordan arrays and gave a recurrence relation for the family of polynomials associated to Riordan arrays. In recent works [16, 17], a new definition by means of a determinant form for Appell polynomials is given. Sequences of Appell polynomials are special of the Sheffer sequences [18]. In [19], the author obtains a determinant representation for the Sheffer sequence. The aim of this work is to propose a similar approach for polynomial sequences of Riordan type, which are special of the generalized Sheffer sequences [12, 18]. A determinant representation for polynomial sequences of Riordan type is obtained by using production matrix of Riordan array. In fact, we will show that the general formula for the polynomial sequences of Riordan type can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, determinant expressions for some classical polynomial sequences such as Fibonacci, Pell, and Chebyshev are derived, and a unified determinant expression for the four kinds of Chebyshev polynomials [20, 21] is established.

2. Main Theorem

In this section we are going to develop our main theorem.

Theorem 4. Let be a Riordan array with the Z-sequence and the A-sequence . Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and . In general, for all , is given by the following Hessenberg determinant:

Proof. Let and . Then from definition and (3), we have and . Hence and . Letting , then and , where . Letting be the production matrix of , then , and . Thus , and . In matrix form, we have
Using the block matrix method, we get
Since The previous matrix equation can be rewritten as Therefore, , and for , we have or equivalently
By applying the Cramer’s rule, we can work out the unknown operating with the first equations in (15):
After transferring the last column to the first position, an operation which introduces the factor , the theorem follows.

Corollary 5. Let be a Riordan array with production matrix . Let be the polynomial sequence associated with . Then , and for all , where , is the principal submatrix of order of the production matrix and is the identity matrix of order .

3. Applications

A useful application of Theorem 4 is to find the determinant expression of a well-known sequence. We illustrate the ideal in the following examples. In the final paragraph, we will give a unified determinant expression for the four kinds of Chebyshev polynomials.

Example 6. Considering the Riordan array , we have . The generating functions of the - and -sequences of are

Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and . In general, is also given by the following Hessenberg determinant: If , , then becomes the Fibonacci polynomials: If , , then gives the Pell polynomials: In case , , becomes the Chebyshev polynomials of the second kind:

Example 7. Considering the Riordan array , we have . Then the generating functions of the - and -sequences of are

Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and , .

In general, is also given by the following Hessenberg determinant:

If , , then become the Chebyshev polynomials of the first kind :

In case , , , give the Fermat polynomials (see [15]):

Example 8. Considering the Riordan array , we have . The generating functions of the - and -sequences of are

Let be the polynomial sequence associated with . Then satisfies the recurrence relation: with initial condition , and , .

In general, is also given by the following Hessenberg determinant:

If , , , then becomes the Chebyshev polynomials of the third kind :

If , , , then gives the Chebyshev polynomials of the fourth kind :

Finally, considering the Riordan array , we have . Then the generating functions of the - and -sequences of are

Let be the polynomial sequence associated with . Then satisfies the recurrence relation:

with initial condition , and , . For , we have

Therefore we can give, now, the following.

Definition 9. The Chebyshev polynomial of degree , denoted by , is defined by where is represented by a Hessenberg determinant of order .
Note that  , , , and . Hence, Definition 9 can be considered as a unified form for the four kinds of Chebyshev polynomials.

Acknowledgments

The authors wish to thank the editor and referee for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11261032) and the Natural Science Foundation of Gansu Province (Grant no. 1010RJZA049).