# Quaternion-Type Catalan Transforms of theρ-Fibonacci and ρ-Lucas Numbers

Authors: Kübra Gül
Journal: Journal of Mathematics
 (2023)

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## 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 toρ-Fibonacci and ρ-Lucas sequences and derive some new sequences by using these quaternion-type Catalan transforms.


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## Body

## 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:(1)H=q:q=q0+q1i+q2j+q3k,qs∈ℝ,s=0,1,2,3,where i,j,k are quaternionic units that satisfy the following rules:(2)i2=j2=k2=ijk=−1,ij=k=−ji,jk=i=−kj,ki=j=−ik.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 ρ any positive real number and p,q any positive integer, the generalized ρ-Fibonacci sequence Gρ,n is defined by(3)Gρ,n=pρFρ,n−1+qFρ,n−2;n≥2,with the initial conditions Gρ,0=a and Gρ,1=b. Some of the particular cases of generalized ρ-Fibonacci sequence are given as follows:If we seta=0,b=1,p=q=1, then Gn is the well-known ρ-Fibonacci sequence,If we seta=2,b=ρ,p=q=1, then Gn is the well-known ρ-Lucas sequence.The generating functions of theρ-Fibonacci sequence Fρ,nn∈ℕ and ρ-Lucas sequence Lρ,nn∈ℕ, respectively, are(4)gx=x1−ρx−x2,hx=2−ρx1−ρx−x2.In [13], the Catalan numbers, with general term Cn, are defined by(5)Cn=1n+12nn.Its generating function is given by(6)cx=1−1−4x2x.The first few Catalan numbers, forn=0,1,2,…, are as follows:(7)1,1,2,5,14,42,132,429,1430,4862,16796,…,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,n≥0, are defined by(8)Cnq=1n+12nnqn,with a quaternion q satisfying one of the followings;(9)q=i,n≡0mod3,q=j,n≡1mod3,q=k,n≡2mod3,where i,j,k are quaternionic units in the (2).
From (8), we can write the quaternion-type Catalan numbers as as follows:(10)Cnq=2n!n+1!n!qn.
The first few terms of this sequence are as follows:(11)n012345678910Cnq1j−2−5i1442k−132−429j14304862i−16796.
It is seen that the relationship between the quaternion-type Catalan numbers and classical Catalan numbers is as follows:(12)Cnq=Cn,forn≡0,4,8mod12,−Cn,forn≡2,6,10mod12,jCn,forn≡1mod12,−iCn,forn≡3mod12,kCn,forn≡5mod12,−jCn,forn≡7mod12,iCn,forn≡9mod12,−kCn,forn≡11mod12.
That is,(13)Cnq2=Cn2,ifnis even,−Cn2,ifnis odd.
Also, we can derive the quaternion-type Catalan numbers as the product of a1×n quaternionic matrix and a n×1 matrix that its elements are Catalan numbers the following matrix relation:(14)i4j5k6i7j8k9⋯=11251442⋮=C0qC1qC2qC3qC4qC5q⋮.
By the (8), we can obtain the following lower-triangular matrix Cq is called as the quaternionic Catalan matrix:(15)Cq=jjk22j2k2i35j5k23i3j414j14k29i34j4k542j42k228i314j45k5i6⋮⋮⋮⋮⋮⋮⋱.
In [13], the terms cm,n of the Catalan matrix C which the first column is the sequence of the Catalan numbers are written as follows:(16)cm,n=∑r=0nnrr2m−1m+r22m−n.
For more information on the matrixC, see [13, 22]. From the definition of the quaternion-type Catalan numbers, it can be easily seen that the term cm,nq of matrix Cq is as follows:(17)cm,nq=qncm,n.
The inverse of the quaternionic Catalan matrix is given by(18)Cq−1=−j1−10−2ii01−3100−3k4k−k001−65−1⋮⋮⋮⋮⋮⋮⋱.
The element of inverse of the quaternionic Catalan matrix is(19)cm,nq−1=−1nqmnm−n.
Suppose thatH is a quaternionic matrix as follows:(20)H=jjk2jk2i3jk2i3j4jk2i3j4k5jk2i3j4k5i6⋮⋮⋮⋮⋮⋮⋱.
Then, the quaternionic Catalan matrixCq can be written as Hadamard product of the Catalan matrix C and the quaternionic matrix H, that is Cq=C∘H. Assume that the terms of the matrices Cq−1 and C−1∘H are denoted by cqm,n−1 and c−1∘hm,n, respectively. Then, we obtain the following equation(21)qmqncqm,n−1=c−1∘hm,n.
It is clearly seen thatqmqn is not equal qm+n.
The generating function of the quaternion-type Catalan numbers is given by(22)cqx=1−1−4qx2qx=1+jx−2x2−5ix3+14x4+⋯.
Now 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 sequenceCnqis given by(23)cqx=12qxexp∑n=1∞1/n1−4qxn.Proof.
Considering MacLaurin expansion for the exponential function, and the equation (10), we obtain the following equation(24)ln2qx.cqx=ln1−1−4qx=−1−4qx−121−4qx2−131−4qx3−⋯=−∑n=1∞1n1−4qxn.
So, we can write(25)cqx=12qxexp∑n=1∞1/n1−4qxn,which is the desired result.Proposition 2.
A combinatorial representation of the quaternion-type Catalan sequenceCnqis as follows:(26)cqx=∑n=0∞12n+12n+1nqnxn.Proof.
By the generalized binomial theorem, it is readily seen that(27)1−4qx=1−124qx−1816q2x2−11643q3x3−512844q4x4−725645q5x5−⋯,1−1−4qx2qx=1−1−2qx−2q2x2−4q3x3−10q4x4−28q5x5−⋯2qx=1+qx+2q2x2+5q3x3+14q4x4+⋯=∑n=0∞12n+12n+1nqnxn.
Thus 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 an. Finally, we derive some new sequences by using the quaternion-type Catalan transform of the ρ-Fibonacci and ρ-Lucas numbers.We define the sequencecqan, which we call the quaternion-type Catalan transform of the sequence an, as follows:(28)cqan=∑m=0nm2n−m2n−mn−mqmam.From the (28), we can write the quaternion-type Catalan transform of the sequence an as the product of the quaternionic Catalan matrix Cq and a n×1 matrix as follows:(29)cqa1cqa2cqa3cqa4cqa5cqa6⋮=jjk22j2k2i35j5k23i3j414j14k29i34j4k542j42k228i314j45k5i6⋮⋮⋮⋮⋮⋮⋱a1a2a3a4a5a6⋮.The elements of the lower triangular matrixCn,n−mq is called as the quaternionic Catalan triangle can be shown by the formula, for 0≤m≤n,(30)Cn,n−mq=2n−m!m+1n−m!n+1!qn−m.It is clearly seen that the inverse of the quaternion-type Catalan transform is the sequencean defined by(31)an=∑m=0n−1mqnmn−mcqam.From the (28), we introduce the quaternion-type Catalan transform of the ρ -Fibonacci sequence as follows:(32)cqFρ,n=∑m=0nm2n−m2n−mn−mqmFρ,m,with cqFρ,0=0.LetAx be the generating function of any sequence an. In [13], by means of cx the generating function of the Catalan numbers, it is proved that Ax∗cx is the generating function of the Catalan transform of the sequence an. Hence, we can write the generating function of the sequence cqFρ,n.(33)CFqx=gx∗cqx=q1−1−4qx−3+q2x−ρ+qρ+11−4qx,where gx is the generating function of the ρ -Fibonacci sequence (3).In this part, taking into account the quaternion-type Catalan transform of theρ-Fibonacci sequence, we give some members of the quaternion-type Catalan transform of the first ρ-Fibonacci numbers. These members of the sequence cqFρ,n are the polynomials in ρ:(34)cqFρ,1=∑m=11m2−m2−m1−mqmFρ,m=j,cqFρ,2=∑m=12m4−m4−m2−mqmFρ,m=−ρ+j,cqFρ,3=∑m=13m6−m6−m3−mqmFρ,m=−iρ2−2ρ+2j−i,cqFρ,4=∑m=14m8−m8−m4−mqmFρ,m=ρ3−3iρ2−3ρ+5j−3i,cqFρ,5=kρ4+4ρ3+3k−9iρ2−6ρ+14j−9i+k,cqFρ,6=−ρ5+5kρ4+10ρ3+−28i+15kρ2−17ρ+42j−28i+5k,cqFρ,7=−jρ6−6ρ5+−5j+20kρ4+24ρ3+−90i−6j+60kρ2−54ρ−90i+131j+20k.Forρ=1,2,3,…, we derive the following quaternion-type Catalan transforms of the ρ-Fibonacci sequence:(35)CqF1=0,j,j−1,2j−2i−2,5j−6i−4,5k+14j−18i−2,…,CqF2=0,j,j−2,2j−5i−4,5j−15i+2,29k+14j−45i+20,…,CqF3=0,j,j−3,2j−10i−6,5j−30i+18,109k+14j−90i+90,….By means of the coefficients of the quaternion-type Catalan transform of theρ-Fibonacci sequence, we derive an infinite triangle, the first few rows of which are given in Table 1.Table 1
The quaternion-type Catalan triangle of theρ-Fibonacci sequence.
cqFρ,1jcqFρ,2−1jcqFρ,3−i−22j−icqFρ,41−3i−35j−3icqFρ,5k43k−9i−614j−9i+kcqFρ,6−15k10−28i+15k−1742j−28+5kcqFρ,7−j−6−5j+20k24−90i−6j+60k−54−90i+131j+20kFrom the above triangle, we can obtain the following results:First diagonal sequencedm=j,j,2j−i,5j−3i,14j−9i+k,42j−28i+5k,−90i+131j+20k,… is the quaternion-type Catalan transform of the sequence 1,0,1,0,1,0,1,…, as A000035, in OEIS. Let cm,iq be the m th row and the i th column element of matrix Cq. The first diagonal sequence, denoted by dm, is given by(36)dm=∑i=1ncm,iq,ifnis odd,∑i=1n−1cm,iq,ifnis even.Second diagonal sequence−1,−2,−3,−6,−17,−54,… as negative values of A319283, in OEIS, is the quaternion-type Catalan transform of the sequence(37)bm=j,1,i,1,2k,2,2j,2,3i,…,defined by(38)bm=−1m/2m4qm,m≥1.A recurrence relation fork≥1 is given by(39)b2kb2k−1=−1k+11q2k−1,b2k+1b2k=−q2k+1,ifkis odd,k+2kq2k+1,ifkis even.Also, we can give the following relations:(40)bmbm−1=−1qm−1,form≡0mod4,1qm−1,form≡2mod4,ifmis even,m−1m−3qm,form≡1mod4,−qm,form≡3mod4,ifmis odd,Note that the sequencecqnam is the n th iteration of the quaternion-type Catalan transform of the sequence am. The sequence cqnam means the n th iteration of the sequence cqam, i.e.,(41)cqnam=cqcqn−1am,n≥1.The second iteration of the quaternion-type Catalan transform of the firstρ-Fibonacci numbers, that means, the first members of the sequence cq2Fρ,n are the polynomials in ρ:(42)cq2Fρ,1=∑m=11m2−m2−m1−mqmcqFρ,m=−1,cq2Fρ,2=∑m=12m4−m4−m2−mqmcqFρ,m=ρ−j−1,cq2Fρ,3=∑m=13m6−m6−m3−mqmcqFρ,m=−ρ2+2i+2ρ−2k−2j−3,cq2Fρ,4=∑m=14m8−m8−m4−mqmcqFρ,m=ρ3+ρ2−3i−3+ρ6i+2−6k−3i−8,cq2Fρ,5=−ρ4+ρ34k+4+ρ2−9j−12i−12+ρ−6k+18i+2−18k−3j−26i−24,cq2Fρ,6=ρ5+ρ4−5k−5+ρ320k+4+ρ2−15k−45j−14i−43+ρ−30k+56i+17−61k−59j−84i−77,cq2Fρ,7=−ρ6+ρ56j+6+ρ4−30k−20i−25+ρ380k−24j−12+ρ2−180k−180j−36i−156+ρ−120k+54j+180i+90−300k−324j−276i−111.From the coefficients of the second iteration of the quaternion-type Catalan transform of theρ-Fibonacci sequence, we derive an infinite triangle given the first few rows in Table 2.Table 2
The triangle of the sequencecq2Fρ,n.
cq2Fρ,1−1cq2Fρ,21−j−1cq2Fρ,3−12i+2−2k−2j−3cq2Fρ,41−3i−36i+2−6k−3i−8cq2Fρ,5−14k+4−9j−12i−12−6k+18i+2−18k−3j−26i−24cq2Fρ,61−5k−520k+4−15k−45j−14i−43−30k+56i+17−61k−59j−84i−77cq2Fρ,7−16j+6−30k−20i−2580k−24j−12−180k−180j−36i−156−120k+54j+180i+90−300−324j−276i−111First diagonal sequence(43)−1,−j−1,−2k−2j−3,−6k−3i−8,−18k−3j−26i−24,−61k−59j−84i−77,−300k−324j−276i−111,…,is the quaternion-type Catalan transform of the sequence(44)j,j,2j−i,5j−3i,14j−9i+k,42j−28i+5k,−90i+131j+20k,…,which is the first diagonal sequence in (36) of the quaternion-type Catalan triangle of the ρ-Fibonacci sequence.In the rest part of the paper, we consider the quaternion-type Catalan transform of theρ-Lucas sequence defined by(45)cqLρ,n=∑m=0nm2n−m2n−mn−mqmLρ,m,with cqLρ,0=0. Then we obtain new sequences from the quaternion-type Catalan transform of the ρ-Lucas sequence.We know thathx and cqx are the generating functions of the ρ-Lucas sequence Lρ,nn∈ℕ and the quaternion-type Catalan sequence Cnq, respectively. It is easy to see that the generating function of the quaternion-type Catalan transform of the ρ-Lucas sequence is as follows:(46)CLqx=hx∗cqx=−4−qρ1−1−4qx−3+q2x−ρ+qρ+11−4qx.Now, we give some members of the quaternion-type Catalan transform of the firstρ-Lucas numbers. These members of the sequence cqLρ,n are the polynomials in ρ:(47)cqLρ,1=∑m=11m2−m2−m1−mqmLρ,m=jρ,cqLρ,2=∑m=12m4−m4−m2−mqmLρ,m=−ρ2+jρ−2,cqLρ,3=∑m=13m6−m6−m3−mqmLρ,m=−iρ3−2ρ2+2j−3iρ−4,cqLρ,4=∑m=14m8−m8−m4−mqmLρ,m=ρ4−3iρ3−ρ2+5j−9iρ−8,cqLρ,5=kρ5+4ρ4+5k−9iρ3+2ρ2+5k+14j−27ρ−20,cqLρ,6=−ρ6+5kρ5+8ρ4+25k−28iρ3+5ρ2+25k+42j−84iρ−58,cqLρ,7=−jρ7−6ρ6+20k−7jρ5+12ρ4+−90i+100k−14jρ3+6ρ2+125j−270i+100kρ−180,cqLρ,8=ρ8−7jρ7−19ρ6+−49j+7kρ5+23ρ4+−98j+35k−297iρ3+4ρ2+35k−891i+380jρ−580.Forρ=1,2,3,…, we derive the following quaternion-type Catalan transforms of the ρ-Lucas sequence:(48)CqL1=0,j,j−3,2j−4i−6,5j−12i−8,11k+14j−9i−43,55k+42j−112i−51,CqL2=0,2j,2j−6,4j−14i−12,10j−42i+4,82k+28j−72i−2,410k+84j−392i+26,…,CqL3=0,3j,3j−11,6j−36i−22,15j−108i+64,393k+42j−243i+231,….By means of the coefficients of the quaternion-type Catalan transform of theρ-Lucas sequence, in Table 3, we derive the following triangle:Table 3
The quaternion-type Catalan triangle of theρ-Lucas sequence.
cqLρ,1jcqLρ,2−1j−2cqLρ,3−i−22j−3i−4cqLρ,41−3i−15j−9i−8cqLρ,5k45k−9i25k+14j−27−20cqLρ,6−15k825k−28i525k+42j−84i−58cqLρ,7−j−620k−7j12−90i+100k−14j6125j−270i+100k−180First diagonal sequence0,−2,−4,−8,−20,−58,−180,−580,… is the quaternion-type Catalan transform of the sequence 0,2,0,2,0,2,0,2,….Second diagonal sequence(49)j,j,2j−3i,5j−9i,5k+14j−27i,25k+42j−84i,125j−270i+100k,375k−891i+380j,…,is the quaternion-type Catalan transform of the sequence 1,0,3,0,5,0,7,0,…, A193356, in OEIS.The second iteration of the quaternion-type Catalan transform of the firstρ-Lucas numbers, that means, the first members of the sequence cq2Lρ,n are the polynomials in ρ:(50)cq2Lρ,1=∑m=11m2−m2−m1−mqmcqLρ,m=−ρ,cq2Lρ,2=∑m=12m4−m4−m2−mqmcqLρ,m=ρ2−j+1ρ−2,cq2Lρ,3=∑m=13m6−m6−m3−mqmcqLρ,m=−ρ3+2i+2ρ2+−2j−2k−5ρ+4+4i,cq2Lρ,4=∑m=14m8−m8−m4−mqmcqLρ,m=ρ4+−3i−3ρ3+6i+4ρ2+−6k−9i−14ρ+12i+2,cq2Lρ,5=−ρ5+4k+4ρ4+−9j−12i−14ρ3+2k+18i+10ρ2+−45k+6j−50i−46ρ−20k+36i−4,cq2Lρ,6=ρ6+−5k−5ρ5+20k+6ρ4+−25k−45j−14i−53ρ3+10k+56i+23ρ2+−216k−14j−112i−151ρ−100k+112i+30,cq2Lρ,7=−ρ7+6j+6ρ6+−30k−20i−27ρ5+80k−12jρ4+−180j−240k−76i−204ρ3+40k+54−6j+180iρ2+−144j−377−1140k−308iρ+360i−400k+228+180j.From the coefficients of the second iteration of the quaternion-type Catalan transform of theρ-Lucas sequence, in Table 4, we derive the following triangle:Table 4
The triangle of the sequencecq2Lρ,n.
cq2Lρ,1−1cq2Lρ,2−1−j−12cq2Lρ,3−12i+2−2j−2k−54+4icq2Lρ,41−3i−36i+4−6k−9i−1412i+2cq2Lρ,5−14k+4−9j−12i−142k+18i+10−45k+6j−50i−46−20k+36i−4cq2Lρ,61−5k−520k+6−25k−45j−14i−5310k+56i+23−216k−14j−112−151−100k+112i+30cq2lρ,7−16j+6−30k−20i−2780k−12j−18−240k−76i−20440k+54−6j+180i−144j−377−1140k−308i−400k+180j+360i+228First diagonal sequence(51)0,2,4i+4,12i+2,−20k+36i−4,−100k+112i+30,360i−400k+228+180j,…,is the quaternion-type Catalan transform of the sequence 0,−2,−4,−8,−20,−58,−180,….Second diagonal sequence(52)−1,−j−1,−2j−2k−5,−6k−9i−14,−45k+6j−50i−46,−216k−14j−112i−151,−144j−377−1140k−308i,…,is the quaternion-type Catalan transform of the sequence(53)j,j,2j−3i,5j−9i,5k+14j−27i,25k+42j−84i,125j−270i+100k,375k−891i+380j,….



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