On the -Central Coefficient Matrices of the Catalan Triangles

Abstract

We introduce the definition of the -central coefficient matrices of a given Riordan array. Applying this definition and Lagrange Inversion Formula, we can calculate the -central coefficient matrices of Catalan triangles and obtain some interesting triangles and sequences.

1. Introduction

Riordan arrays have drawn the attention of various authors from many points of view in the recent literature. We recall some concepts and properties of Riordan arrays. An infinite lower triangular matrix is called a Riordan array if the generating function of its column is , where and are formal power series with , , and . We also write ; the general term of matrix is , where is the coefficient operator.

The set of all Riordan arrays forms a group under ordinary matrix multiplication. The group law is then given by The identity is and the inverse of is where is compositional inverse of .

If we multiply the Riordan array by a column vector with generating function , then we get a column vector whose generating function is given by . The rule can be rewritten as which is the Fundamental Theorem of the Riordan group.

Lemma 1 (see [1]). An infinite lower triangular matrix is a Riordan array if and only if there exists a sequence such that Such a sequence is called the A-sequence of the Riordan array . Besides, the A-sequence is uniquely determined by the function according to the following formula and vice versa:

In the calculation below we need the Lagrange Inversion Formula and some knowledge of generalized binomial series.

Lemma 2 ((LIF) [2]). Supposing that a formal power series is implicitly defined by the relation , where is a formal power series such that ; then for any formal power series , where stands for the generating function for ; that is, .

Lemma 3 (see [2, 3]). For any integer , the generalized binomial series is given by for which we have

Given a Riordan array , then its central coefficient matrix is defined as . For example, the central coefficient matrix of the Pascal matrix , which is one of the most classical matrices, plays an important role in combinatorics. The first few rows of its central coefficient matrix [4] are In [5], this matrix is used to give a new proof of an identity of Andrews [6]. In this paper, we describe a process of obtaining new Riordan matrices from a given Riordan array, which corresponds to taking the central coefficient matrix several times. By considering the Catalan triangles, we obtain some interesting triangles and sequences. The first Catalan triangle we consider is with , which was introduced by Aigner [7]. The first few rows are where is the generation function of the Catalan numbers. The second Catalan triangle is with , , which was introduced by Shapiro [8]. The first few rows are The third Catalan triangle is with , , which was introduced by Radoux [9]. The first few rows are

2. The -Central Coefficient Matrices of the Catalan Triangles

Definition 4. Let be a Riordan array; then we say that are the -central coefficient matrices of , where and .

According to the definition, the 0-central coefficient matrix is itself; the 1-central coefficient matrix is equal to central coefficient matrix. In genaral, the -central coefficient matrix is the central coefficient matrix of . For example, the first few rows of the 2-central coefficient matrix of Pascal matrix are

Theorem 5. The -central coefficient matrices of can be written as Moreover the generating functions of the A-sequences of are .

Proof. On the one hand, for any , , we have By (9) and (10), we have On the other hand, let , which implies ; by (5), we finish the proof.

Theorem 6. The inverse Riordan arrays of can be written as

Proof. Let . Then . We have known that ; then and . Hence the result follows by (2).

The first few rows of and are

Theorem 7. The -central coefficient matrices of can be written as and the generation functions of the A-sequences of are .

Proof. On the one hand, for any , , we have On the other hand, let , which implies ; by (5) we finish the proof.

Theorem 8. The inverse Riordan arrays of can be written as

Proof. Let . Then . Since , then and . Hence the result follows by (2).

The first few rows of and are

Theorem 9. The -central coefficient matrices of can be written as and the generation functions of the A-sequences of are .

Proof. For any , , we have Hence we finish the proof.

Theorem 10. The inverse Riordan arrays of can be written as

Proof. This can be easily obtained from the proof of Theorem 8.
The first few rows of and are

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11261032).