# Catalan Transform ofk-Balancing Sequences

Authors: Asim Patra; Mohammed K. A. Kaabar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In this work, the Catalan transformation (CT) ofk-balancing sequences, Bk,nn≥0, is introduced. Furthermore, the obtained Catalan transformation CBk,nn≥0 was shown as the product of lower triangular matrices called Catalan matrices and the matrix of k-balancing sequences, Bk,nn≥0, which is an n×1 matrix. Apart from that, the Hankel transform is applied further to calculate the determinant of the matrices formed from CBk,nn≥0.


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## Body

## 1. Introduction

A balancing numbern is a positive integer that satisfies the Diophantine equation as follows:(1)1+2+⋯+n−1=n+1+⋯+n+r,for some positive integer r, called the balancer corresponding to n [1]. The n-th balancing number is denoted by Bn. For each n, 8Bn2+1 is a perfect square, and its positive square root is called a Lucas-balancing number [2]. The balancing numbers are solutions of Pell’s equation y2−8x2=1, and its Binet formula is given by(2)Bn=α2n−β2nα2−β2,where α=1+2 and β=1−2. Furthermore, the balancing numbers Bn satisfy the recurrence relation as follows:(3)Bn+1=6Bn−Bn−1,with initial terms B0=0 and B1=1 [2–10].There are several transformations that operate on various integer sequences. Some of them are Binomial Transform [11], Discrete Cosine Transform (DCT) [12], Laguerre Transform [13], Lah Transform [14], Discrete Wavelet Transform (DWT) [15], Discrete Fourier Transform (DFT), and Catalan Transform [16]. In the real life processes, these transformations are basically utilized in various Steganographic schemes and techniques [12–14]. In addition, there are numerous other usages of the above displayed transforms but despite that certain shortcomings of some of the transform techniques are still observed. For instance, the coefficients of the DCT are real-valued which makes the calculation very much time-consuming. Furthermore, the coefficients of DFT are, in general, complex numbers which generate complex value as output. In addition, the computational complexity of DFT is On2 which also suggests that the transform is cumbersome and time-consuming. To overcome the shortcomings discussed regarding the different transformations, the Catalan transformation, introduced by Paul Barry [16] in 2005, proves to be a superior alternative. The reason behind CT being an appropriate substitute is due to the fact that unlike the earlier mentioned transformations, the overall calculation of Catalan transform is integer-based and not based on floating-point values which makes it very faster and reliable method of transformation. Furthermore, the computational complexity of Catalan transform is Onlogn which also indicates a substantial improvement as we go from On2 of DFT. Apart from the real life phenomena, the beneficial property of the Catalan transform makes it quite suitable to be applied for the derivation of important results relating to the properties of several integer sequences [17–19]. Several classical core sequences, like the Fibonacci sequence Fn [20], Pell sequence Pn [21], and Jacobsthal sequence Jn [17], can be paired by means of this transformation.There are some research studies where the Catalan transform has been applied to thek-Fibonacci sequences [22]. Furthermore, the Catalan transform was applied to the k-Jacobsthal sequence in [23]. In [24], the Catalan transformation was applied to the k-Lucas sequence. In addition, a new sequence realizing the known Lucas numbers has been discussed by Özkan et al. [25]. In application point of view, Mukhopadhyay et al. [26] dealt with the secured image steganography via CT. In addition, the Lah transform has been applied for security data in the field of telecommunication in [14]. The past literature depicts the Catalan transform of various sequences but it was observed that the balancing sequence was never been tackled via the Catalan transform which illustrates the novelty of the present work. Hence, in the present work, deriving motivation from the literature survey, we employed the Catalan transformation in case of the k-balancing sequence Bk,nn≥0, discussed in the later section, and inspected the properties of the sequences. Then, we apply the Hankel transform to the Catalan transform of Bk,n.The paper has been arranged in the following manner: first the preliminaries related to thek-balancing sequence, Catalan numbers, Hankel transform, and Catalan transform are presented in Section 2. Furthermore, Section 3 is devoted towards the evaluation of Catalan transformation of the k-balancing sequence. Apart from that, the generating function is calculated for the CT of k-balancing sequence in Section 4. The next section deals with the Hankel transform of CT of k-balancing numbers. The concluding remarks are mentioned in the final section.


## 2. Preliminaries

### 2.1.k-Balancing Numbers

Thek-balancing numbers, denoted by Bk,nn=0∞, for any positive number k, are defined by the recursive relation as follows [27]:(4)Bk,n+1=6kBk,n−Bk,n−1,n≥1,with the initial values Bk,0=0 and Bk,1=1. Furthermore, the Binet formula for k-balancing numbers is given by(5)Bk,n=α1n−α2nα1−α2,where α1=3k+9k2−1 and α2=3k−9k2−1. Hence, the first few k-balancing numbers are given by Bk,0=0, Bk,1=1, Bk,2=6k, Bk,3=36k2−1, Bk,4=216k3−12k, Bk,5=1296k4−108k2+1, and so on. The k-balancing numbers are tabulated in Table 1.Table 1
Somek-balancing numbers.
nBk,n001126k336k2−14216k3−12k51296k4−108k2+167776k5−864k3+18k


### 2.2. Catalan Numbers

The Catalan numbers [28], termed after the French-Belgian mathematician named Euge`ne Charles Catalan, are defined as a sequence of natural numbers that are encountered in several counting problems and specifically occur where the items are defined recursively. They are denoted by Cn and are further defined by [16], in terms of binomial coefficients, as(6)Cn=1n+12nn=2n!n+1!n!.Furthermore, by virtue of [29], we have the recurrence relation of the Catalan numbers Cn as(7)Cn+1=22n+1n+2Cn,with C0=1, and its ordinary generating function is given by [29](8)cx=1−1−4x2x,(9)=1+x+2x2+5x3+14x4+⋯.Furthermore, for somen, the first few Catalan numbers [28] are given by (sequence A000108 in [30]):(10)1,1,2,5,14,42,132,429,1430,4862,16796,….


### 2.3. The Hankel Transform

By definition, the Hankel matrix of a sequence of real numbersg0,g1,g2,… is given by the infinite matrix as follows [31]:(11)ℋng=g0g1g2g3g4g5…g1g2g3g4g5g6…g2g3g4g5g6g7…g3g4g5g6g7g8…g4g5g6g7g8g9…⋮⋮⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋱.Furthermore, the Hankel transform [29] of a sequence of real numbers g0,g1,… is defined as the sequence of determinants as follows:(12)Hn=Detℋng,where ℋng is the Hankel matrix given by ℋng=gi+j−20<i,j≤n+1. Alternatively, it can be said that the Hankel transform produces the sequence of determinants of the Hankel matrices formed from the known or given sequence. Furthermore, the Hankel transform of the Catalan sequence is given by the sequence 1,1,1,… as given in [30]. In addition, the Hankel transform of the sum of consecutive generalized Catalan numbers is the bisection of the classical Fibonacci sequence [32].Lemma 1.
The Hankel transform of a sequence is invariant under the binomial transform of that sequence. In other words, if we write the binomial transform of a given sequencexn as(13)yn=∑k=0nnkxk,then(14)Det=xi+j−21≤i,j≤n+1=yi+j−21≤i,j≤n+1.Proof.
The proof of the lemma can be referred from [31].


### 2.4. The Catalan Transformation

Definition 1.
The Catalan transformation, as introduced in [16], is a sequence transform which is defined as follows.
Let us considerdnn≥0 to be a sequence with generating function as follows:(15)Gx=d0+d1x+d2x2+⋯.
Then, the Catalan transformation ofdn is defined to be the sequence whose ordinary generating function (o.g.f.) is given by Gxcx, where cx is the series as defined in (8) [16]. The Catalan transformation is also linked to various known transforms, particularly the binomial transformation.Lemma 2.
The Catalan transformationbn of a given sequence an is given by(16)bn=∑i=0ni2n−i2n−in−iai.Proof.
The proof of the lemma can be referred from [16].


## 2.1.k-Balancing Numbers

Thek-balancing numbers, denoted by Bk,nn=0∞, for any positive number k, are defined by the recursive relation as follows [27]:(4)Bk,n+1=6kBk,n−Bk,n−1,n≥1,with the initial values Bk,0=0 and Bk,1=1. Furthermore, the Binet formula for k-balancing numbers is given by(5)Bk,n=α1n−α2nα1−α2,where α1=3k+9k2−1 and α2=3k−9k2−1. Hence, the first few k-balancing numbers are given by Bk,0=0, Bk,1=1, Bk,2=6k, Bk,3=36k2−1, Bk,4=216k3−12k, Bk,5=1296k4−108k2+1, and so on. The k-balancing numbers are tabulated in Table 1.Table 1
Somek-balancing numbers.
nBk,n001126k336k2−14216k3−12k51296k4−108k2+167776k5−864k3+18k


## 2.2. Catalan Numbers

The Catalan numbers [28], termed after the French-Belgian mathematician named Euge`ne Charles Catalan, are defined as a sequence of natural numbers that are encountered in several counting problems and specifically occur where the items are defined recursively. They are denoted by Cn and are further defined by [16], in terms of binomial coefficients, as(6)Cn=1n+12nn=2n!n+1!n!.Furthermore, by virtue of [29], we have the recurrence relation of the Catalan numbers Cn as(7)Cn+1=22n+1n+2Cn,with C0=1, and its ordinary generating function is given by [29](8)cx=1−1−4x2x,(9)=1+x+2x2+5x3+14x4+⋯.Furthermore, for somen, the first few Catalan numbers [28] are given by (sequence A000108 in [30]):(10)1,1,2,5,14,42,132,429,1430,4862,16796,….


## 2.3. The Hankel Transform

By definition, the Hankel matrix of a sequence of real numbersg0,g1,g2,… is given by the infinite matrix as follows [31]:(11)ℋng=g0g1g2g3g4g5…g1g2g3g4g5g6…g2g3g4g5g6g7…g3g4g5g6g7g8…g4g5g6g7g8g9…⋮⋮⋮⋮⋮⋮⋱⋮⋮⋮⋮⋮⋮⋱.Furthermore, the Hankel transform [29] of a sequence of real numbers g0,g1,… is defined as the sequence of determinants as follows:(12)Hn=Detℋng,where ℋng is the Hankel matrix given by ℋng=gi+j−20<i,j≤n+1. Alternatively, it can be said that the Hankel transform produces the sequence of determinants of the Hankel matrices formed from the known or given sequence. Furthermore, the Hankel transform of the Catalan sequence is given by the sequence 1,1,1,… as given in [30]. In addition, the Hankel transform of the sum of consecutive generalized Catalan numbers is the bisection of the classical Fibonacci sequence [32].Lemma 1.
The Hankel transform of a sequence is invariant under the binomial transform of that sequence. In other words, if we write the binomial transform of a given sequencexn as(13)yn=∑k=0nnkxk,then(14)Det=xi+j−21≤i,j≤n+1=yi+j−21≤i,j≤n+1.Proof.
The proof of the lemma can be referred from [31].


## 2.4. The Catalan Transformation

Definition 1.
The Catalan transformation, as introduced in [16], is a sequence transform which is defined as follows.
Let us considerdnn≥0 to be a sequence with generating function as follows:(15)Gx=d0+d1x+d2x2+⋯.
Then, the Catalan transformation ofdn is defined to be the sequence whose ordinary generating function (o.g.f.) is given by Gxcx, where cx is the series as defined in (8) [16]. The Catalan transformation is also linked to various known transforms, particularly the binomial transformation.Lemma 2.
The Catalan transformationbn of a given sequence an is given by(16)bn=∑i=0ni2n−i2n−in−iai.Proof.
The proof of the lemma can be referred from [16].


## 3. Catalan Transformation of thek-Balancing Sequence

By virtue of Lemma2, we define the Catalan transformation of the k-balancing sequence, Bk,nn≥0, as(17)CBk,n=∑i=0ni2n−i2n−in−iBk,i,with CBk,0=0. Hence, it can be observed that the members of Catalan transform of the first k-balancing numbers are the polynomials in k given by(18)CBk,1=∑i=01i2−i2−i1−iBk,i=1,CBk,2=∑i=02i4−i4−i2−iBk,i=1+6k,CBk,3=∑i=03i6−i6−i3−iBk,i=1+12k+36k2,CBk,4=∑i=04i8−i8−i4−iBk,i=2+18k+108k2+216k3=5+56k+336k2−1+216k3−12k,CBk,5=∑i=05i10−i10−i5−iBk,i=6+36k+216k2+864k3+1296k4=14+146k+936k2−1+4216k3−12k+1296k4−108k2+1,CBk,6=∑i=06i12−i12−i6−iBk,i=19+102k+468k2+2160k3+6480k4+7776k5=42+426k+2836k2−1+14216k3−12k+51296k4−108k2+1+18k−864k3+7776k5.Furthermore, we can write the above equations as the product of a lower triangular matrix sayC and a n×1 matrix Bk which is depicted as(19)CBk,1CBk,2CBk,3CBk,4CBk,5...=111221553114149414242281451............Bk,1Bk,2Bk,3Bk,4Bk,5....The elements of matrixC satisfy the recursive relation given by(20)Ca,b=∑k=b−1a−1Ca−1,k.Furthermore, the first and second column ofC are equal for a>1, denoting the Catalan numbers. In addition to it, the lower triangular matrix denoted by Cn,n−a is known as the Catalan triangle and its entries satisfy the relation as follows [22]:(21)Cn,n−a=a+12n−a!n+1!n−a!,0≤a≤n.Hence, the Catalan triangle is obtained via the coefficients of the Catalan transform of thek-balancing sequence and is given by the figure in Table 2.Table 2
Catalan triangle of thek-balancing sequence.
CB11CB261CB336121CB4216108182CB51296864216366CB677766480216046810219…………………In addition, we obtain the first few terms of the series of Catalan transform of thek-balancing sequence as follows:(22)CB1=0,1,7,49,344,2418,17005,119617…,CB2=0,1,13,169,2198,28590,371887,4837381,…,CB3=0,1,19,361,6860,130362,2477305,47076937,…,CB4=0,1,25,625,15626,390678,9767659,244209229,…,CB5=0,1,31,961,29792,923586,28632229,887632081,…,CB6=0,1,37,1369,50654,1874238,69348295,2565942037,….


## 4. Generating Function of the Catalan Transform ofk-Balancing Sequence

The generating functions (g.f.s), initially introduced by Abraham de Moivre, are a step to encode an infinite sequence of numbers via treating them as the coefficients of a power series. In other words, the problems about sequences can be transformed into problems about functions by means of the generating functions. Now, in this case, we find the generating function of the Catalan transformation of thek-balancing sequence and denote it by CBkx. By virtue of (8), we have the generating function of the Catalan numbers as cx. Furthermore, by virtue of [16], it can be clearly observed that Zcx is the generating function of the Catalan transform of any sequence zn. Hence, by means of the g.f. of the k-balancing sequence in [27], we have(23)CBkx=Bkx.cx=x.cx1−6kx.cx−x.cx2=1−1−4x1+2x−6k+6k+11−4x.


## 5. Hankel Transform of Catalan Transform ofk-Balancing Numbers

In the present section, we find the Hankel transform of the Catalan transform ofk-balancing numbers in the form of determinants of Hankel matrices. By considering the Catalan transform of the k-balancing sequence of the preceding section, we obtain the following:(24)HCB1=DetCBk,1=Det1=1=1,HCB2=CBk,1CBk,2CBk,2CBk,3=11+6k1+6k1+12k+36k2=0,HCB3=CBk,1CBk,2CBk,3CBk,2CBk,3CBk,4CBk,3CBk,4CBk,5=11+6k1+12k+36k21+6k1+12k+36k22+18k+108k2+216k31+12k+36k22+18k+108k2+216k36+36k+216k2+864k3+1296k4=−1,HCB4=CBk,1CBk,2CBk,3CBk,4CBk,2CBk,3CBk,4CBk,5CBk,3CBk,4CBk,5CBk,6CBk,4CBk,5CBk,6CBk,7=11+6k1+12k+36k27+6k+108k2+216k31+6k1+12k+36k22+18k+108k2+216k36+36k+216k2+864k3+1296k41+12k+36k22+18k+108k2+216k36+36k+216k2+864k3+1296k419+102k+468k2+2160k3+6480k4+7776k5=−1.


## 6. Conclusion

In the present study, the Catalan transformation is applied to thek-balancing sequence and some identities are obtained. The identities were further represented in matrix form, and the terms of the CT of k-balancing sequence are displayed. Apart from that, the Catalan triangle is obtained by means of the coefficients of the Catalan transform of the k-balancing sequence. In addition, the generating function of CT of the sequence is manifested. Finally, the Hankel transform is utilized to the Catalan transformation of the known k-balancing sequence.



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