Abstract
In this paper, we define a new sequence called the quaternion-type Catalan sequence and give generating function, exponential representation, quaternionic Catalan matrix and its some properties. Then, we introduce a quaternion-type Catalan transform, which is a generalization of the Catalan transform, and its inverse. Moreover, we apply the quaternion-type Catalan transforms to -Fibonacci and -Lucas sequences and derive some new sequences by using these quaternion-type Catalan transforms.
1. Introduction
Quaternions were invented by Irish mathematician W. R. Hamilton (1805–1865) as an extension to the complex numbers. Quaternion algebra recently has played a significant role in several areas of science such as the quantum physics, computer science, analysis and differential geometry [1, 2].
In [3], Hamilton introduced the set of quaternions which can be represented as follows:where are quaternionic units that satisfy the following rules:
The number sequences have many applications in quaternion theory. The study of the quaternions of the sequences began with the earlier work of Horadam [4] where the Fibonacci quaternion was investigated. In the literature, there are several studies on different quaternions and their generalizations; for example, [5–9].
Many authors introduced some generalizations of Fibonacci numbers, e.g. [4, 10–12]. In [12], for any positive real number and any positive integer, the generalized -Fibonacci sequence is defined bywith the initial conditions and . Some of the particular cases of generalized -Fibonacci sequence are given as follows: If we set , then is the well-known -Fibonacci sequence, If we set , then is the well-known -Lucas sequence.
The generating functions of the -Fibonacci sequence and -Lucas sequence , respectively, are
In [13], the Catalan numbers, with general term , are defined by
Its generating function is given by
The first few Catalan numbers, for , are as follows:cited in [14], from now on OEIS, as A000108.
In the literature, there are several studies on the Catalan numbers and their generalizations, see [15–21]. Deveci and Shannon [18] mentioned an attractive definition, namely the complex-type Catalan numbers. Here we introduce a new generalization of the Catalan numbers that we call the quaternion-type Catalan numbers. Then we give some of its properties, such as the generating matrix, the generating function, the exponential representation and combinatorial representation.
2. Quaternion-Type Catalan Numbers
Definition 1. The quaternion-type Catalan numbers, , are defined bywith a quaternion satisfying one of the followings;where are quaternionic units in the (2).
From (8), we can write the quaternion-type Catalan numbers as as follows:The first few terms of this sequence are as follows:It is seen that the relationship between the quaternion-type Catalan numbers and classical Catalan numbers is as follows:That is,Also, we can derive the quaternion-type Catalan numbers as the product of a quaternionic matrix and a matrix that its elements are Catalan numbers the following matrix relation:By the (8), we can obtain the following lower-triangular matrix is called as the quaternionic Catalan matrix:In [13], the terms of the Catalan matrix which the first column is the sequence of the Catalan numbers are written as follows:For more information on the matrix , see [13, 22]. From the definition of the quaternion-type Catalan numbers, it can be easily seen that the term of matrix is as follows:The inverse of the quaternionic Catalan matrix is given byThe element of inverse of the quaternionic Catalan matrix isSuppose that is a quaternionic matrix as follows:Then, the quaternionic Catalan matrix can be written as Hadamard product of the Catalan matrix and the quaternionic matrix that is . Assume that the terms of the matrices and are denoted by and , respectively. Then, we obtain the following equationIt is clearly seen that is not equal .
The generating function of the quaternion-type Catalan numbers is given byNow we give exponential and combinatorial representations for the quaternion-type Catalan sequence by using the generating function with the following propositions.
Proposition 1. An exponential representation of the quaternion-type Catalan sequence is given by
Proof. Considering MacLaurin expansion for the exponential function, and the equation (10), we obtain the following equationSo, we can writewhich is the desired result.
Proposition 2. A combinatorial representation of the quaternion-type Catalan sequence is as follows:
Proof. By the generalized binomial theorem, it is readily seen thatThus the proof is completed.
3. Quaternion-Type Catalan Transforms
In [13], the author introduced the Catalan transform of a given sequence and its inverse in terms of the Riordan group, and then derive many transformed sequences. By inspired of this work, the Catalan transform has been applied to many sequences; for example [18, 22–25]. In this section, we define the quaternion-type Catalan transform and then give the generating function of the quaternion-type Catalan transform of a given sequence . Finally, we derive some new sequences by using the quaternion-type Catalan transform of the -Fibonacci and -Lucas numbers.
We define the sequence , which we call the quaternion-type Catalan transform of the sequence , as follows:
From the (28), we can write the quaternion-type Catalan transform of the sequence as the product of the quaternionic Catalan matrix and a matrix as follows:
The elements of the lower triangular matrix is called as the quaternionic Catalan triangle can be shown by the formula, for ,
It is clearly seen that the inverse of the quaternion-type Catalan transform is the sequence defined by
From the (28), we introduce the quaternion-type Catalan transform of the -Fibonacci sequence as follows:with .
Let be the generating function of any sequence . In [13], by means of the generating function of the Catalan numbers, it is proved that is the generating function of the Catalan transform of the sequence . Hence, we can write the generating function of the sequence where is the generating function of the -Fibonacci sequence (3).
In this part, taking into account the quaternion-type Catalan transform of the -Fibonacci sequence, we give some members of the quaternion-type Catalan transform of the first -Fibonacci numbers. These members of the sequence are the polynomials in :
For , we derive the following quaternion-type Catalan transforms of the -Fibonacci sequence:
By means of the coefficients of the quaternion-type Catalan transform of the -Fibonacci sequence, we derive an infinite triangle, the first few rows of which are given in Table 1.
From the above triangle, we can obtain the following results:
First diagonal sequence is the quaternion-type Catalan transform of the sequence , as A000035, in OEIS. Let be the th row and the th column element of matrix . The first diagonal sequence, denoted by , is given by
Second diagonal sequence as negative values of A319283, in OEIS, is the quaternion-type Catalan transform of the sequencedefined by
A recurrence relation for is given by
Also, we can give the following relations:
Note that the sequence is the th iteration of the quaternion-type Catalan transform of the sequence . The sequence means the th iteration of the sequence , i.e.,
The second iteration of the quaternion-type Catalan transform of the first -Fibonacci numbers, that means, the first members of the sequence are the polynomials in :
From the coefficients of the second iteration of the quaternion-type Catalan transform of the -Fibonacci sequence, we derive an infinite triangle given the first few rows in Table 2.
First diagonal sequenceis the quaternion-type Catalan transform of the sequencewhich is the first diagonal sequence in (36) of the quaternion-type Catalan triangle of the -Fibonacci sequence.
In the rest part of the paper, we consider the quaternion-type Catalan transform of the -Lucas sequence defined bywith . Then we obtain new sequences from the quaternion-type Catalan transform of the -Lucas sequence.
We know that and are the generating functions of the -Lucas sequence and the quaternion-type Catalan sequence , respectively. It is easy to see that the generating function of the quaternion-type Catalan transform of the -Lucas sequence is as follows:
Now, we give some members of the quaternion-type Catalan transform of the first -Lucas numbers. These members of the sequence are the polynomials in :
For , we derive the following quaternion-type Catalan transforms of the -Lucas sequence:
By means of the coefficients of the quaternion-type Catalan transform of the -Lucas sequence, in Table 3, we derive the following triangle:
First diagonal sequence is the quaternion-type Catalan transform of the sequence .
Second diagonal sequenceis the quaternion-type Catalan transform of the sequence , A193356, in OEIS.
The second iteration of the quaternion-type Catalan transform of the first -Lucas numbers, that means, the first members of the sequence are the polynomials in :
From the coefficients of the second iteration of the quaternion-type Catalan transform of the -Lucas sequence, in Table 4, we derive the following triangle:
First diagonal sequenceis the quaternion-type Catalan transform of the sequence .
Second diagonal sequenceis the quaternion-type Catalan transform of the sequence
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
The author received no financial support for the research authorship and publication of this article.