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International Journal of Combinatorics
Volume 2013 (2013), Article ID 528584, 6 pages
http://dx.doi.org/10.1155/2013/528584
Research Article

Some Inverse Relations Determined by Catalan Matrices

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 24 June 2013; Revised 21 August 2013; Accepted 21 August 2013

Academic Editor: Laszlo A. Szekely

Copyright © 2013 Sheng-liang Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use the -sequence and -sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients. Some matrix identities obtained by Shapiro and Radoux are all special cases of our identity. In addition, a unified form of Catalan matrices is introduced.

1. Introduction

The Catalan numbers have been widely encountered and investigated [1, 2]. They can be defined through binomial coefficients or by the generating function being which satisfies the functional equation . In [2], Stanley listed 66 enumerative problems which are counted by the Catalan numbers. Many number triangles related to the Catalan sequence have been introduced in the literature. In [35], Shapiro et al. introduced a Catalan triangle with the entries given by The following identity is obtained in [6] in connection with the moment of the Catalan triangle: Another proof of the above identity is given by Woan et al. [7] while computing the areas of parallelo-polyominoes via generating functions. In [8], a combinatorial interpretation of the matrix identity (4) is also obtained.

In [9], Radoux introduced a triangle of numbers and he presents the identity with , which is equivalent to following matrix equation: Deng and Yan [10] proved this identity by using the Riordan array method.

Aigner [11] introduced a number triangle with the entries given by This array is also discussed in [1214].

We use the -sequence and -sequence of Riordan array to characterize the inverse relation associated with the Riordan array. We apply this result to prove some combinatorial identities involving Catalan matrices and binomial coefficients, which are generalizations of (4) and (6). In addition, a unified form of Catalan matrices is introduced.

2. Riordan Arrays

In the recent literature, one may find that Riordan arrays have attracted the attention of various authors from many points of view, and many examples and applications can be found (see, e.g., [13, 1521]). An infinite lower triangular matrix is called a Riordan array if its column has generating function , where and are formal power series with , , and . The Riordan array is denoted by . Thus, the general term of Riordan array is given by where denotes the coefficient of in power series . Suppose we multiply the array by a column vector and get a column vector . Let be the ordinary generating function for the sequence . Then it follows that the ordinary generating function for the sequence is . If we identify a sequence with its ordinary generating function, the composition rule can be rewritten as This is called the fundamental theorem for Riordan arrays, and this leads to the multiplication rule for the Riordan arrays: The set of all Riordan arrays forms a group under ordinary multiplication. The identity is . The inverse of is where is compositional inverse of .

Lemma 1 (see [22, 23]). Let be an infinite lower triangular matrix. Then is a Riordan array if and only if and there exist two sequences and with and such that Such sequences are called the -sequence and the -sequence of the Riordan array , respectively.

Lemma 2 (see [22, 23]). Let be a Riordan array, and let and be the generating functions for the corresponding - and -sequences, respectively. Then we have If the inverse of is . Then

Example 3. (a) It is well known that the Pascal matrix can be expressed as the Riordan array , and the generating functions of its - and -sequences are . More generally, it is easy to show that the generalized Pascal array can be expressed as the Riordan matrix and the generating functions of its - and -sequences are , and .
(b) For nonnegative integer , the Pascal functional -eliminated matrix was introduced in [24] by . We have , and the generating functions of its - and -sequences are , . We also have .

Definition 4. 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 to the Riordan array .

If is the polynomial sequence associated to a Riordan array , and let be its generating function, then by (9), we have Thus, . The notion of the polynomial sequence of Riordan type was introduced in [25], and it has been studied by [26]. In this paper, by studying the polynomial sequence of Riordan type related to some Catalan type matrices, we obtain some interesting identities and inverse relations.

Theorem 5. Let be a Riordan array, and let and be the generating functions of its -sequence and -sequence. If , then where is any real number.

Proof. Let ; then .

Corollary 6. Let be a Riordan array. If and are the generating functions of its -sequence and -sequence, respectively, and if is the polynomial sequence associated to the Riordan array , then

Proof. By the theorem, we have Hence, The result then follows from identity (15).

Corollary 7. Let be a Riordan array. If and are the generating functions of its -sequence and -sequence, respectively, and if is the polynomial sequence associated to the Riordan array , then where is the -element of . In matrix form, we have

Example 8. The lower triangular matrix in (4) may be represented as The generating functions of its - and -sequence are . Because , from Theorem 5, we have This is exactly the matrix identity (4).

Example 9. The lower triangular matrix in (6) can be written as The generating functions of its - and -sequence are . Since , from Theorem 5, we have This is exactly the matrix identity (6).

3. Inverse Relations Determined by Catalan Matrices

Let , be integer numbers, and let be arbitrary parameter. We define the generalized Catalan matrix to be the Riordan array where is the generating function of Catalan sequence defined in (2). From [27], we have for any integer number . Hence . Therefore, by (8), the generic element of the generalized Catalan matrix is given by Denote . For , , the corresponding matrices are widely studied by many authors [5, 12, 14, 15, 28, 29]. For example, , , , and . The matrix is the Catalan triangle introduced by Aigner [11] and studied in [6, 12, 13]. The matrix is the Catalan triangle defined by Shapiro [5]; see also (4). The matrix is the Catalan matrix defined by Radoux [9]; see also (6).

Theorem 10. Let be the polynomial sequence associated to the Riordan array . Then, the identities hold for every .

Proof. From generic term given in (27) with and , we have the generic term of Catalan matrix which is , and by simplifying we obtain . Using (11) and (14), we get , and the generating functions of its - and -sequences are and . By Corollary 6, . Therefore, , , and , for . Solving this recurrence relation, we have . From Corollary 7, we get the results.

For the case , , we have . Thus, . By Theorem 10, we obtain The last identity is equivalent to the following matrix identity:

Theorem 11. Let be the polynomial sequence associated to the Riordan array . Then, for any nonnegative integer , one has

Proof. From generic term given in (27) with and , we have the generic term of Catalan matrix which is . Using (11), we obtain . Hence, the generic term of is . By Corollary 7, we obtain the desired results.

Since , from Lemma 2, the generating functions of - and -sequences of are and . Hence, . For the case and , we have , and . By Theorem 11, we have The last identity is equivalent to identity (6).

For the case and , we have , and , where is Fibonacci sequence with generating . By Theorem 11, we have

Theorem 12. Let be the polynomial sequence associated to the Riordan array . Then, one has for every .

Proof. From generic term given in (27) with and , we have the generic term of Catalan matrix which is . Using (11), we obtain . Hence, the generic term of is . By Corollary 7, we obtain the desired results.

The generating functions of - and -sequences of are and . Thus .

For the case and , we have , where are Chebyshev polynomials of the second kind (see [1]). Hence, by Theorem 13, we have Substituting in the last identity, we get (4) again.

Theorem 13. Let be the polynomial sequence associated to the Riordan array . Then, one has identities

Proof. Using (11), we obtain . Hence, the generic term of is . From generic term given in (27) with and , we have the generic term of Catalan matrix which is , and by simplifying we obtain . From Corollary 7, we obtain the desired results.

The generating functions for the - and -sequences of are , and . For the case and , we have . By Theorem 13, we have The matrix form of the last identity is

Acknowledgments

The author would like to thank the referees for their helpful suggestions. This work was supported by the National Natural Science Foundation of China (11261032).

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