Ann. Acad. Rom. Sci.  
Ser. Math. Appl.  
ISSN 2066-6594  
Vol. 18, No. 2/2026  
NEW PROPERTIES AND IDENTITIES FOR  
BICOMPLEX FIBONACCI FINITE OPERATOR  
SEQUENCES∗  
Istayan Das†  
Binod Chandra Tripathy‡  
Mausumi Sen§  
Communicated by G. Moro¸sanu  
DOI  
10.56082/annalsarscimath.2026.2.241  
Abstract  
In this paper, we construct a new class of bicomplex sequences  
whose components are Fibonacci finite operator sequences. A sys-  
tematic study of their structural properties is carried out within the  
framework of the idempotent decomposition of bicomplex numbers.  
We derive explicit recurrence relations, polynomial representations,  
and matrix formulations for these bicomplex Fibonacci finite opera-  
tor sequences. Several identities are established using the associated  
matrix representation.  
Keywords: bicomplex number, Fibonacci number, finite operator, Fi-  
bonacci polynomial, matrix representation.  
MSC: 11B37, 11B39, 11B83, 05A15.  
1 Introduction and preliminaries  
In recent years, the theory of bicomplex numbers has become an evolving  
area of mathematical research, leading to substantial developments and the  
∗Accepted for publication on March 10, 2026  
†dasistayan95@gmail.com, Department of Mathematics, National Institute of Tech-  
nology Silchar, Silchar 788010, Assam, India  
‡tripathybc@gmail.com, Department of Mathematics, Tripura University, Suryaman-  
inagar, Agartala 799022, Tripura, India  
§mausumi@math.nits.ac.in, Department of Mathematics, National Institute of Tech-  
nology Silchar, Silchar 788010, Assam, India  
241  
On bicomplex Fibonacci finite operator sequences  
242  
emergence of new research directions. In the historical development of bi-  
complex numbers, Segre [17] is credited with their initial introduction. Sub-  
sequently, Price [15] carried out a systematic study of derivatives, integrals,  
and their higher-dimensional generalizations in the bicomplex setting.  
A bicomplex number is defined as  
z = z1 + jz2 = x + iy + ju + ijv,  
where z1 = x + iy and z2 = u + iv, with z1, z2 ∈ C and x, y, u, v ∈ R. Here,  
R and C denote the sets of real and complex numbers, respectively. The  
imaginary units i and j satisfy  
i2 = j2 = −1,  
ij = ji.  
We denote the set of bicomplex numbers by BC. The algebra of BC  
possesses two distinguished idempotent zero divisors  
1 + ij  
1 − ij  
e1 =  
,
e2 =  
.
2
2
Every bicomplex number z = z1 + jz2 can be uniquely expressed in the  
idempotent form  
z = β1e1 + β2e2,  
where β1 = z1 − iz2 and β2 = z1 + iz2. This representation is known as the  
idempotent decomposition of a bicomplex number.  
For z = β1e1 + β2e2 and w = γ1e1 + γ2e2, we have  
z ± w = (β1 ± γ1)e1 + (β2 ± γ2)e2,  
zw = (β1γ1)e1 + (β2γ2)e2.  
The Euclidean norm on BC is defined by  
p
2
2
kzkBC  
=
=
|z1| + |z2|  
r
2
2
|β1| + |β2|  
.
2
The theory of bicomplex numbers has been further developed and applied  
by several authors including Beg and Dutta [1], Sager and Sa˘gır [16]. Various  
classes of bicomplex sequence spaces have been introduced and studied by  
Srivastava and Srivastava [18], Wagh [22], and Bera and Tripathy [2–4], and  
many others.  
I. Das, B.C. Tripathy, M. Sen  
243  
(1)  
The Fibonacci sequence {Fn} is defined by the recurrence relation  
Fn+2 = Fn+1 + Fn,  
n ≥ 0,  
with initial conditions F0 = 0 and F1 = 1.  
The Binet formula for the Fibonacci sequence is given by  
αn − βn  
√
Fn =  
,
5
√
2
√
2
1+  
5
1−  
5
where α =  
and β =  
.
The Fibonacci polynomial Fn(x) defined by  
1,  
n = 0,  
n = 1,  
Fn+1(x) =  
x,  
xFn(x) + Fn−1(x), n ≥ 2.  
For the details of Fibonacci Number and Fibonacci polynomial, one may  
refer to Koshy [13], Vajda [20], Verner and Hoggatt [21], Dunlap [6], and  
many others.  
Kızılate¸s [8] introduced the concept of the Fibonacci finite operator se-  
(k)  
ꢀ
quence and established its Binet-type representation. Let F = Fn  
de-  
note the k-th Fibonacci finite operator sequence defined by the recurrence  
relation  
(k)  
(k)  
Fn+1 = Fn(k) + Fn−1  
,
(2)  
where n, k ≥ 1.  
The corresponding Binet-type representation of the sequence is given by  
(k)  
0
Fn = F1(k)Fn + F  
F
,
(3)  
(k)  
n−1  
where {Fn} denotes the classical Fibonacci sequence. For the Fibonacci  
finite operator sequence one may refer to [8] and the refer ences therein.  
The bicomplex Fibonacci number [7] is defined by  
BCFn = Fn + iFn+1 + jFn+2 + ijFn+3  
Its Binet-type formula is  
.
(4)  
(5)  
n
n
ˆ
αˆα − ββ  
√
BCFn =  
,
5
     
On bicomplex Fibonacci finite operator sequences  
244  
where  
2
3
αˆ = 1 + iα + jα2 + ijα3,  
ˆ
β = 1 + iβ + jβ + ijβ .  
The idempotent representation of BCFn is given by  
¯
BCFn = βne1 + βne2,  
0
0
¯
where βn = zn + izn, βn = zn − izn,  
zn = Fn + iFn+1  
,
z0 = Fn+2 + iFn+3  
.
n
For further details of bicomplex Fibonacci numbers and their further  
generalizations one may refer to [5,9,10,14] and the references therein.  
The extension of classical Fibonacci-type sequences to hypercomplex  
number systems has attracted considerable attention. Various generaliza-  
tions have been obtained by embedding Fibonacci numbers into quater-  
nionic, octonionic, sedenionic, and bicomplex frameworks, enriching both  
algebraic and structural properties.  
Kızılate¸s and Kone [10] developed higher-order Fibonacci 2m-ions and  
derived recurrence relations, Binet formulas, generating functions, and clas-  
sical identities. Kızılate¸s [11] introduced incomplete Fibonacci and Lucas  
quaternions and obtained their fundamental properties.  
In the bicomplex setting. In the context of hypercomplex generaliza-  
tions, Kızılate¸s et. al. [12] introduced higher-order Fibonacci quaternions  
using higher-order Fibonacci numbers, extending the classical quaternionic  
framework studied by A. F. Horadam and S¸. Halıcı. They established recur-  
rence relations, Binet formulas, generating functions, and several algebraic  
properties.  
Halıcı [7] introduced bicomplex Fibonacci numbers and derived Binet-  
type formulas and matrix representations. Kızılate¸s et al. [9] studied general-  
ized bicomplex Tribonacci quaternions. Terzio˘glu et al. [19] investigated Fi-  
bonacci finite operator quaternions and established several identities. More-  
over, C¸u¨ru¨k and Halıcı [5] studied bicomplex Fibonacci numbers via idem-  
potent decomposition.  
Despite these developments, most works analyze bicomplex Fibonacci-  
type sequences component-wise. A systematic investigation of bicomplex  
Fibonacci finite operator sequences explicitly formulated through the idem-  
potent representation has not yet been fully developed.  
Motivated by this observation, the present paper establishes new recur-  
rence relations, polynomial representations, and matrix formulations of bi-  
complex Fibonacci finite operator sequences via the idempotent basis. This  
I. Das, B.C. Tripathy, M. Sen  
245  
approach simplifies derivations and reveals structural identities that are not  
immediately visible in the classical representation.  
2 Main results  
Throughout this section, we work entirely in the bicomplex idempotent  
framework. The idempotent decomposition enables a component-wise anal-  
ysis of bicomplex Fibonacci finite operator sequences, thereby simplifying  
the derivation of structural properties. This approach provides a systematic  
method to obtain recurrence relations, polynomial expressions, and matrix  
representations, and reveals identities that are not directly observable in the  
standard bicomplex form.  
2.1  
Algebraic properties of bicomplex Fibonacci finite  
operators and their new attributes  
In this section we study bicomplex Fibonacci finite operator sequences and  
established some of their recurrence relations.  
Definition 1. For each n ≥ 0, the bicomplex Fibonacci finite operator  
(k)  
BCFn is defined by  
(k)  
(k)  
(k)  
(k)  
BCFn = Fn(k) + i Fn+1 + j Fn+2 + ij Fn+3  
,
(6)  
(k)  
where Fn denotes the k-th finite operator sequence.  
The bicomplex Fibonacci finite operator can be written as  
BCFn = Fn(k) + u,  
(k)  
(k)  
(k)  
(k)  
where u = iFn+1 + jFn+2 + ijFn+3  
.
(k)  
The conjugate of the bicomplex Fibonacci finite operator, BCFn is denoted  
ꢁ ꢂ∗  
(k)  
by BCFn  
where  
ꢁ ꢂ∗  
(k)  
(k)  
BCFn  
= Fn − u.  
(7)  
For the bicomplex Fibonacci finite operator, we may obtain  
ꢁ ꢂ∗  
(k)  
(k)  
BCFn(k) + BCFn  
= 2Fn  
.
Remark 1. The idempotent representation of bicomplex Fibonacci finite  
operator sequence is given as  
BCFn = βn(k)e1 + β(k)e2,  
(k)  
¯
n
On bicomplex Fibonacci finite operator sequences  
246  
(k)  
(k)  
k
where βn = pn − ip0n, βnk = pn + ip0n, pn = Fn + iFn+1 and p0n  
=
¯
(k)  
(k)  
F
n+2 + iFn+3  
Theorem 1. For each integer n ≥ 2, the bicomplex Fibonacci finite operator  
(k)  
BCFn admits the representation  
ꢃ ꢄ ꢃ ꢄ  
(k)  
(k)  
(k)  
n−1  
(k)  
n−2  
(k)  
¯
¯
BCFn  
=
βn−1 + βn−2 e1 +  
β
+ β  
e2.  
(8)  
Proof. Using equation (2) we have  
(k)  
(k)  
(k)  
(k)  
BCFn =Fn(k) + iFn+1 + jFn+2 + ijFn+3  
ꢁ ꢂ ꢁ ꢂ ꢁ ꢂ  
(k)  
(k)  
(k)  
(k)  
= Fn−1 + Fn−2 + i Fn(k) + Fn−1 + j Fn+1 + Fn  
(k)  
ꢁ ꢂ  
(k)  
(k)  
+ ij Fn+2 + Fn+1  
ꢁ
ꢂ
(k)  
(k)  
(k)  
= Fn−1 + iFn(k) + jFn+1 + ijFn+2  
ꢁ
ꢂ
(k)  
(k)  
(k)  
+ Fn−2 + iFn−1 + jFn(k) + ijFn+1  
(k)  
(k)  
=BCFn−1 + BCFn−2  
ꢃ ꢄ ꢃ ꢄ  
(k)  
(k)  
(k)  
n−1  
(k)  
n−2  
¯
¯
= βn−1 + βn−2 e1 +  
β
+ β  
e2.  
Remark 2. The above expression also can be written as  
ꢃ ꢄ ꢃ ꢄ  
(k)  
(k)  
(k)  
n−1  
(k)  
n−2  
βn(k)e1 + β(k)e2 = βn−1 + βn−2 e1 +  
β
+ β  
e2.  
(9)  
¯
¯
¯
n
(k)  
Theorem 2. Let {BCFn  
}
be the bicomplex Fibonacci finite operator  
n≥0  
sequence. Then, for each n ≥ 1, the sequence admits the following Binet-  
type representation:  
ꢅ ꢆ ꢅ ꢆ  
(k)  
(k)  
αˆ αn−1 αF1(k) + F0  
− β β  
5
βF1(k) + F0  
n−1  
ˆ
(k)  
√
BCFn  
=
,
(10)  
where α and β are the roots of the characteristic equation associated with  
the corresponding Fibonacci-type recurrence.  
     
I. Das, B.C. Tripathy, M. Sen  
247  
Proof. Using the equation (5) and (3) we have  
(k)  
(k)  
(k)  
BCFn =Fn(k) + iFn+1 + jFn+2 + ijFn+3  
(k)  
ꢁ ꢂ  
(k)  
0
(k)  
=F1(k)Fn + F  
F
+ i F  
F
+ F(k)Fn  
n−1  
n+1  
1
0
ꢁ
ꢂ ꢁ  
ꢂ
(k)  
(k)  
(k)  
(k)  
0
+ j F1  
F
+ F  
F
+ ij F1  
F
+ F  
F
n+2  
n+2  
n+1  
n+3  
1
=F1(k)(Fn + iFn+1 + jFn+2 + ijFn+3  
)
+ F0(k)(Fn−1 + iFn + jFn+1 + ijFn+2  
)
=F1(k)BCFn + F(k)BCFn−1  
0
n
n
n−1  
(k) αˆαn−1 − ββ  
ˆ
ˆ
(k) αˆα − ββ  
√
√
=F1  
+ F0  
5
5
ꢃ ꢄ ꢃ ꢄ  
αˆαn−1 αF1(k) + F0  
− ββ  
βF1(k) + F0  
(k)  
(k)  
n−1  
ˆ
√
=
.
5
(k)  
Theorem 3. Let {BCFn  
}
be the bicomplex Fibonacci finite operator  
n≥0  
sequence. Then its ordinary generating function  
∞
X
BCF(k)(z) =  
BCFn(k)zn  
n=0  
is given by  
h
i
(β0(k)e1 + β(k)e2) + (β(k)e1 + β(k)e2) − (β(k)e1 + β(k)e2) z  
¯
¯
¯
0
1
1
0
0
BCF(k)(z) =  
.
1 − z − z2  
Proof. Let BCFn(k)(z) be the generating function of the bicomplex Fibonacci  
finite operator, defined as  
∞
X
BCFn(k)(z) =  
BCFn(k)zn.  
n=0  
Then, we have,  
BCFn(k)(z) = BCF0(k) + BCF1(k)z + BCF2 z2 + · · · + BCFn(k)zn + · · ·  
(k)  
(k)  
(k)  
− zBCFn(k)(z) = −BCF0(k)z − BCF1 z2 − BCF2 z3 − · · · − BCFn(k)zn+1 − · · ·  
(k)  
(k)  
(k)  
− z2BCFn(k)(z) = −BCF0 z2 − BCF1 z3 − BCF2 z4 − · · · − BCFn(k)zn+2 − · · ·  
 
On bicomplex Fibonacci finite operator sequences  
248  
Using the above identities and equation (8), we have  
ꢀ
ꢁ
ꢀ
ꢁ
(k)  
(k)  
(k)  
1 − z − z2 BCFn(k)(z) =BCF0  
+
BCF1 − BCF0  
z+  
ꢀ
ꢁ
∞
X
(k)  
(k)  
(k)  
BCFn − BCFn−1 − BCFn−2 zn  
n=2  
ꢀ
ꢁ
ꢂ
ꢃ
1 − z − z2 BCFn (z) =(β0(k)e1 + β(k)e2) + (β1(k)e1 + β(k)e2) − (β(k)e1 + β(k)e2) z  
(k)  
¯
¯
¯
0
0
1
0
∞
ꢂ
X
(k)  
(k)  
(βn(k)e1 + βn(k)e2) − (βn−1e1 + βn(k−)1e2) − (βn−2e1  
¯
¯
¯
+
n=2  
ꢃ
(k)  
n−2  
+ β  
e2) zn  
Following equation (9), we have  
h
i
(β0(k)e1 + β(k)e2) + (β(k)e1 + β(k)e2) − (β(k)e1 + β(k)e2) z  
¯
¯
¯
0
1
1
0
0
BCF(k)(z) =  
.
1 − z − z2  
Theorem 4. The exponential generating function associated with the bi-  
(k)  
complex Fibonacci finite operator sequence {BCFn  
}
is given by  
n≥0  
ꢅ ꢆ ꢅ ꢆ  
(k)  
(k)  
βz  
βz  
F1  
αˆeαz − βe  
− F0  
αˆβeαz − βαe  
ˆ
ˆ
∞
(k) zn  
X
√
BCFn  
=
,
n!  
5
n=0  
√
√
5
1+  
5
1−  
where α =  
and β =  
.
2
2
Proof. Using equation (10) we have  
!
ꢁ ꢂ ꢁ ꢂ  
(k)  
(k)  
αˆαn−1 αF1(k) + F0  
− ββ  
βF1(k) + F0  
∞
∞
n−1  
n
ˆ
X X  
(k) z  
zn  
√
BCFn  
=
=
n!  
n!  
5
n=0  
n=0  
!
ꢁ ꢂ ꢁ ꢂ  
(k)  
(k)  
αˆ αF1(k) + F0  
β βF1(k) + F0  
ˆ
1
eαz  
−
eβz  
√
α
β
5
ꢁ ꢂ ꢁ ꢂ  
(k)  
(k)  
αˆβeαz αF1(k) + F0  
− αˆβeαz βF1(k) + F0  
√
=
=
5αβ  
ꢁ ꢂ ꢁ ꢂ  
(k)  
(k)  
βz  
βz  
ˆ
ˆ
F1  
αˆeαz − βe  
− F0  
αˆβeαz − βαe  
√
.
5
I. Das, B.C. Tripathy, M. Sen  
249  
Theorem 5. Let n be a non-negative integer and let p ∈ Z. Then the  
following identity holds:  
ꢃ ꢄ  
n
ꢅ ꢆ  
X
n
t
(k)  
(k)  
(−1)t  
BCF2t+p = (−1)n βn+pe1 + β(k) e2  
.
¯
n+p  
t=0  
Proof. Observing equation (10) we find that  
!
ꢁ ꢂ  
ꢃ ꢄ  
ꢃ ꢄ  
(k)  
αˆα2t+p−1 αF1(k) + F0  
n
n
X
X
n
n
(k)  
BCF2t+p  
(−1)t  
=
=
(−1)t  
√
t
t
5
t=0  
t=0  
!
ꢁ ꢂ  
ꢃ ꢄ  
(k)  
βF1(k) + F0  
n
2t+p−1  
ˆ
X
ββ  
n
t
√
−
(−1)t  
5
t=0  
ꢁ ꢂ  
αˆαp−1 αF1(k) + F0  
(k)  
ꢁ ꢂ  
n
1 − α2  
√
5
ꢁ ꢂ  
βF1(k) + F0  
(k)  
p−1  
ˆ
ꢁ ꢂ  
ββ  
n
√
−
1 − β2  
5
ꢃ ꢄ  
αˆαp−1 αF1(k) + F0  
(k)  
√
=
(−α)n  
5
ꢃ ꢄ  
βF1(k) + F0  
(k)  
p−1  
ˆ
ββ  
√
−
(−β)n  
5
(k)  
=(−1)nBCFn+p  
.
=(−1) (βn+pe1 + β(k) e2).  
(k)  
n
¯
n+p  
Theorem 6. Let n ∈ N0. Then  
ꢃ ꢄ  
n
X
n
(k)  
= β2(kn)e1 + β(k)e2.  
¯
BCFt  
2n  
t
t=0  
On bicomplex Fibonacci finite operator sequences  
250  
Proof. From equation (10) we have  
  !  
  !  
!
ꢄ
ꢅ
ꢄ
ꢅ
n
n
(k)  
(k)  
αˆαt−1 αF1(k) + F0  
− ββ  
βF1(k) + F0  
t−1  
ˆ
X
X
n
n
(k)  
BCFt  
√
=
t
t
5
t=0  
t=0  
ꢄ
ꢄ
αˆα−1 αF1(k) + F0  
)
ββ  
βF1(k) + F0  
)
(k)  
(k)  
−1  
ˆ
=
=
(1 + α)n  
−
(1 + β)n  
√
5
√
5
ꢄ
ꢄ
αˆα2n−1 αF1(k) + F0  
)
ββ  
βF1(k) + F0  
)
(k)  
(k)  
2n−1  
ˆ
√
√
−
5
5
(k)  
=BCF2n  
=β2(kn)e1 + β2(kn)e2.  
¯
2.2  
Polynomial representation of bicomplex Fibonacci finite  
operators and their new attributes  
In this section we introduce bicomplex Fibonacci polynomial and established  
bicomplex version of some well known inequalities.  
Definition 2. The bicomplex Fibonacci polynomial for finite operator se-  
quences denoted as  
1  
n = 0,  
BCFn(k)(z) and defined as BCFn+1(z) =  
(k)  
z
n = 1,  
zBCFn(k)(z) + BCFn−1(z) n ≥ 2.  
(k)  
The first few bicomplex Fibonacci finite operator polynomials are  
BCF0(k)(z) = 0  
BCF1(k)(z) = 1  
BCF2(k)(z) = z  
(k)  
BCF3 (z) = z2 + 1  
(k)  
BCF4 (z) = z3 + 2z  
(k)  
BCF5 (z) = z4 + 3z2 + 1  
(k)  
BCF6 (z) = z5 + 4z3 + 3z.  
(k)  
We define a new class of matrix Q(z) to study the BCHn+1(z) polynomial  
where,  
ꢇ ꢈ  
z
1
Q(z) =  
1 0  
I. Das, B.C. Tripathy, M. Sen  
251  
Then  
"
#
BCFn+1(z) BCFn(k)(z)  
(k)  
Qn(z) =  
(11)  
BCFn(k)(z) BCFn−1(z)  
(k)  
where n ≥ 1. Since |Q| = −1, |Qn| = (−1)n. Accordingly, equation (11)  
gives the Cassini-like formula for BCFn(k)(z):  
ꢅ ꢆ  
2
(k)  
(k)  
BCFn+1(z)BCFn−1(z) − BCFn(k)(z)  
= (−1)n  
(12)  
It follows from equation (11) that  
"
#
(k)  
BCFm+n+1(z)  
(k)  
BCFm+n(z)  
Qm+n(z) =  
(k)  
BCFm+n(z)  
(k)  
BCFm+n−1(z)  
m+n  
Q
(z)  
m
=Q (z)Q (z)  
n
(k)  
(k)  
(k)  
m
(k)  
m−1  
(k)  
(k)  
(k)  
(k)  
m
(k)  
m−1  
(k)  
n−1  
BCF  
(z)BCF  
n+1  
(z) + BCF (z)BCF (z) BCF  
(z)BCF (z) + BCF (z)BCF  
(z)  
n
n
m+1  
(k)  
BCF (z)BCF  
m+1  
(k)  
(z)BCF (z) BCF (z)BCF (z) + BCF  
=
(k)  
n+1  
(k)  
n
(k)  
(k)  
n−1  
(z) + BCF  
(z)BCF  
(z)  
m
m
n
Consequently,  
(k)  
(k)  
(k)  
BCFm+n(z) = BCFm+1(z)BCFn(k)(z) + BCFm(k)(z)BCFn−1(z)  
(13)  
In particular, let z = 1. This yields an identity  
(k)  
(k)  
(k)  
BCFm+n = BCFm+1BCFn(k) + BCFm(k)BCFn−1  
.
The above identity is known as bicomplex version of Honsberger’s identity.  
Since BCH0(k)(z) = BCF1(k)(z) + BCF0(k)(z) = 1 + 0 = 1 and BCH1(k)(z) =  
BCF2(k)(z) + BCF1(k)(z) = z + 1, it follows that BCHn(k)(z) = BCFn+1(z) +  
(k)  
BCFn(k)(z);  
(k)  
(k)  
∴ (−1)i+1BCHi (z) =(−1)i+1BCFi+1(z) − (−1)iBCFi(k)(z)  
n
n
X
X
(−1)i+1BCHi(k)(z) =  
[(−1)i+1BCFi+1(z) − (−1)iBCFi(k)(z)]  
(k)  
i=0  
i=0  
(k)  
i+1  
=(−1)n+1BCF  
(z)  
Thus we can express the bicomplex Fibonacci polynomial BCFn(k)(z) in  
terms of BCHn(k)(z) polynomial as  
n
X
(k)  
BCFn+1(z) =  
(−1)n+iBCHi(k)(z)  
i=0  
   
On bicomplex Fibonacci finite operator sequences  
252  
Now its follows from the equation (11) and (12)  
"
(k)  
#
BCHn+1(z)  
BCHn(k)(z)  
Qn+1(z) + Qn(z) =  
(k)  
BCHn(k)(z)  
BCHn−1(z)  
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
BCHn(k)(z) ꢉ  
(k)  
BCHn+1(z)  
ꢉ ꢉ  
ꢉ
n
ꢉ ꢉ  
= Q (z)[Q(z) + I]  
ꢉ
(k)  
BCHn−1(z)  
BCHn(k)(z)  
ꢉ
=|Qn(z)| · |Q + I|.  
That is,  
ꢅ ꢆ  
2
(k)  
(k)  
BCHn+1(z)BCHn−1(z) − BCHn(k)(z)  
= z(−1)−n  
(14)  
The above identity is again a generalization of Cassini’s rule in the bicomplex  
setting. We have a further generalization of equation (14):  
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ ꢉ  
ꢉ ꢉ  
ꢉ ꢉ  
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
(k)  
BCHn+a(z) BCH  
(k)  
n+a+b  
(k)  
BCHn+b(z)  
(z)  
BCH(k)(z) BCHa+b(z)  
(k)  
a
= (−1)n  
.
ꢉ ꢉ  
BCHn(k)(z)  
BCH0(k)(z) BCHb (z)  
(k)  
ꢉ ꢉ  
(k)  
Remark 3. If we replace BCFn(z) by (βn e1 + βnke2)(z), then we get the  
k
¯
idempotent representation of bicomplex polynomial.  
2.3  
Matrix representations of bicomplex Fibonacci finite  
operators and their new attributes  
In this section we study matrix representation of bicomplex Fibonacci finite  
operator sequences and established some useful theorems.  
Definition 3. The bicomplex Fibonacci matrix is defined by  
ꢇ ꢈ  
BCF2 BCF1  
P =  
,
(15)  
(16)  
BCF1 BCF0  
and satisfies  
where  
ꢇ ꢈ  
BCFn+1  
BCFn  
BCFn  
BCFn−1  
PQn−1  
=
,
ꢇ ꢈ ꢇ ꢈ  
1 1  
1 0  
Fn+1  
Fn  
Fn  
Q =  
,
Qn =  
.
Fn−1  
 
I. Das, B.C. Tripathy, M. Sen  
253  
From the recurrence relation of the Fibonacci finite operator numbers,  
we can easily see the matrix relation  
" # " #  
(k)  
(k)  
(k)  
Fn+1  
(k)  
F2  
F1  
Fn  
H =  
=⇒ HQn−1  
=
.
(17)  
(k)  
(k)  
(k)  
(k)  
Fn−1  
F1  
F0  
Fn  
Considering the matrix equalities in equation (16) and (17), we have a matrix  
representation of the bicomplex Fibonacci finite operator as follows:  
" #  
(k)  
(k)  
ꢁ ꢂ ꢁ ꢂ  
BCFn+2 BCFn+1  
HQn−1 P = P HQn−1  
=
.
(18)  
(k)  
BCFn+1  
(k)  
BCFn  
Let us note that equality equation (18) is held even though matrix multipli-  
cation is not commutative. Namely  
" #  
ꢇ ꢈ  
(k)  
Fn+1  
(k)  
ꢁ ꢂ  
Fn  
BCF2 BCF1  
HQn−1 P =  
(k)  
(k)  
Fn−1  
BCF1 BCF0  
Fn  
"
#
#
(k)  
(k)  
Fn+1BCF + Fn(k)BCF1 Fn+1BCF + Fn(k)BCF0  
2
Fn(k)BCF2 + F  
(k)  
1
=
=
=
(k)  
n−1  
(k)  
BCF1 Fn(k)BCF1 + Fn−1BCF0  
"
(k)  
(k)  
(k)  
BCF2Fn+1 + BCF1Fn  
BCF1Fn+1 + BCF0Fn  
BCF Fn(k) + BCF1F  
BCF Fn(k) + BCF0F  
(k)  
n−1  
(k)  
n−1  
2
1
" #  
ꢇ ꢈ  
(k)  
(k)  
BCF2 BCF1  
Fn+1  
Fn  
(k)  
(k)  
Fn−1  
BCF1 BCF0  
Fn  
ꢁ ꢂ  
=P HQn−1  
.
Additionally, the Bicomplex Fibonacci finite operator has the following ma-  
trix representation:  
" #  
" #  
(k)  
(k)  
(k)  
(k)  
BCF2  
BCF1  
BCFn+2 BCFn+1  
L :=  
=⇒ QnL = LQn =  
. (19)  
(k)  
(k)  
(k)  
BCFn+1  
(k)  
BCF1  
BCF0  
BCFn  
Equation (19) can be expressed as  
"
#
β2(k)e1 + β(k)e2 β1(k)e1 + β(k)e2  
¯
¯
2
1
L :=  
β1(k)e1 + β(k)e2 β0(k)e1 + β(k)e2  
¯
¯
1
0
and  
"
#
(k)  
(k)  
βn+2e1 + β(k) e2 βn+1e1 + β(k) e2  
¯
¯
QnL = LQn =  
.
n+2  
n+1  
(k)  
βn+1e1 + β(k) e2  
βn(k)e + βn(k)e2  
¯
¯
1
n+1  
       
On bicomplex Fibonacci finite operator sequences  
254  
Theorem 7. For n ≥ 0, we have  
ꢀ
ꢄ
ꢅ
2
(k)  
(k)  
(k)  
¯
¯
0
BCFn+1BCFn−1 − BCFn  
= (−1)n−1 (β2(k)e1 + β(k)e2)(β(k)e1 + β(k)e2)  
2
0
ꢁ
− (β1(k)e1 + β(k)e2)2  
,
(20)  
(21)  
¯
1
ꢀ
ꢄ
ꢅ
2
= (−1)n−1 (β0(k)e1 + β(k)e2)(β(k)e1 + β(k)e2)  
(k)  
(k)  
(k)  
¯
¯
2
BCFn−1BCFn+1 − BCFn  
0
2
ꢁ
ꢄ
ꢅ
2
− β1(k)e1 + β(k)e2  
,
¯
1
ꢄ
ꢅ
ꢅ
ꢄ
2
2
(k)  
(k)  
= (−1)n−1 (β2e1 + β2e2)(β0e1 + β0e2)  
(k)  
¯
¯
BCFn+1BCFn−1 − BCFn  
ꢂ
ꢃ
ꢃ
ꢅ
(k)  
(k) (k)  
(k)  
2
2
2
− (β1e1 + β1e2)  
(F1 )2 − F1 F0 − (F0  
)
)
,
.
(22)  
(23)  
¯
ꢄ
ꢄ
(k)  
(k)  
= (−1)n−1 (β0e1 + β0e2)(β2e1 + β2e2)  
(k)  
¯
¯
BCFn−1BCFn+1 − BCFn  
ꢂ
ꢅ
(k)  
(k) (k)  
(k)  
− (β1e1 + β1e2)  
(F1 )2 − F1 F0 − (F0  
2
¯
Proof. To prove equations (20) and (21), using the equation (19), we get  
ꢉ ꢉ  
ꢉ
(k)  
(k) ꢉ  
ꢉ ꢉ  
BCF  
BCF  
ꢉ ꢉ  
n
(k)  
n−1  
n+1  
(k)  
ꢉ ꢉ  
= Q  
L
ꢉ ꢉ  
ꢉ ꢉ  
BCFn  
BCFn−1  
ꢉ ꢉ  
ꢉ ꢉ  
n−1  
ꢉ
(k)  
(k)ꢉ  
ꢉ ꢉ  
1 1  
BCF  
BCF  
ꢉ ꢉ  
n−1  
2
1
ꢉ ꢉ  
= |Q|  
|L| =  
ꢉ ꢉ  
(k)  
(k)  
ꢉ ꢉ  
1 0  
ꢉ ꢉ  
BCF1  
BCF0  
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ
ꢉ ꢉ  
n−1  
¯
¯
ꢉ ꢉ  
1 1  
β
(k)e1 + β(k)e2 β1(k)e1 + β(k)e2ꢉ  
ꢉ
2
2
1
ꢉ ꢉ  
=
ꢉ
ꢉ
β1(k)e1 + β(k)e2 β0(k)e1 + β(k)e2  
ꢉ ꢉ  
¯
¯
1 0  
1
0
ꢀ
ꢄ
ꢅ
2
BCFn+1BCFn−1 − BCFn  
=(−1)n−1 (β2(k)e1 + β(k)e2)(β(k)e1 + β(k)e2)  
(k)  
(k)  
(k)  
¯
¯
0
2
0
ꢁ
− (β1(k)e1 + β(k)e2)2  
,
¯
1
and  
ꢀ
ꢄ
ꢅ
2
(k)  
(k)  
(k)  
¯
¯
2
BCFn−1BCFn+1 − BCFn  
=(−1)n−1 (β0(k)e1 + β(k)e2)(β(k)e1 + β(k)e2)  
0
2
ꢁ
ꢄ
ꢅ
2
− β1(k)e1 + β(k)e2  
.
¯
1
In the same way, equations (22) and (23) are obtained if we take determi-  
nants on both sides of the matrix equation (18).  
     
I. Das, B.C. Tripathy, M. Sen  
255  
Theorem 8. For integer m, n ≥ 1, the following equalities hold:  
(k)  
(k)  
Fn(k)BCFm+1 + Fn−1BCFm = βm+ne1 + βm(k+) ne2,  
(24)  
(25)  
¯
(k)  
(k)  
FnBCFm+1 + Fn−1BCFm = βm+ne1 + βm(k+) ne2,  
(k)  
¯
BCFm+1BCFn+1 + BCFm(k)BCFn = (β2e1 + β2e2)(βm+ne1 + βm(k+) ne2)  
(k)  
(k)  
¯
¯
+ (β1e1 + (β1e2)(βm+n−1e1 + βm(k+) n−1e2),  
(k)  
(26)  
¯
¯
BCFm+1BCFn+1 + BCFm(k)BCFn = (β2(k)e1 + β(k)e2)(βm+ne1 + βm(k+) ne2)  
(k)  
(k)  
(k)  
(k)  
¯
¯
2
+ (β1(k)e1 + β(k)e2)(βm+n−1e1 + βm(k+) n−1e2). (27)  
(k)  
¯
¯
1
Proof. Substituting n −→ m + n − 1 into (18) and (19), we have  
"
"
#
#
#
(k)  
(k)  
ꢄ
ꢅ
ꢄ
ꢅꢄ  
ꢅ
BCFm+n+1  
BCFm+n  
= HQm+n−2 P = HQn−1 Qm−1  
P
(k)  
BCFm+n  
(k)  
(k)  
BCFm+n−1  
(k)  
"
"
#
#
ꢆ
ꢆ
ꢇ
ꢇ
(k)  
Fn+1  
(k)  
BCFm+n+1  
BCFm+n  
Fn  
BCFm+1  
BCFm  
=
=
(k)  
BCFm+n  
(k)  
BCFm+n−1  
(k)  
(k)  
Fn−1  
(k)  
BCFm  
BCFm−1  
Fn  
"
(k)  
(k)  
(k)  
Fn+1  
βm+n+1e1 + βm(k+) n+1e2  
βm+ne1 + βm(k+) ne2  
Fn  
BCFm+1  
BCFm  
BCFm−1  
¯
¯
.
(k)  
(k)  
(k)  
(k)  
Fn−1  
βm+ne1 + βm(k+) ne2  
βm+n−1e1 + βm(k+) n−1e2  
Fn  
¯
¯
BCFm  
The equation (24) is obtained by comparing the corresponding entries of  
the two matrix equations. From equation (19) we see that  
"
#
#
#
(k)  
BCFm+n+1  
(k)  
BCFm+n  
=Qm+n−1L = Qn−1(QmL)  
(k)  
BCFm+n  
(k)  
(k)  
BCFm+n−1  
(k)  
"
"
"
#
#
ꢆ
ꢆ
ꢇ
ꢇ
(k)  
(k)  
BCFm+n+1  
BCFm+n  
Fn  
Fn−1  
BCFm+2 BCFm+1  
=
=
(k)  
BCFm+n  
(k)  
BCFm+n−1  
(k)  
BCFm+1  
(k)  
(k)  
BCFm  
(k)  
Fn−1 Fn−2  
"
(k)  
βm+n+1e1 + βm(k+) n+1e2  
(k)  
βm+ne1 + βm(k+) ne2  
Fn  
Fn−1 Fn−2  
Fn−1  
BCFm+2 BCFm+1  
¯
¯
.
βm+n−1e1 + βm(k+) n−1e2  
(k)  
(k)  
(k)  
BCFm+1  
(k)  
BCFm  
βm+ne1 + βm(k+) ne2  
¯
¯
The equation (25) is obtained by comparing the corresponding entries of  
the two matrix equations. Likewise, Substituting n −→ m + n − 2 into  
equations (18) and (19) we have  
P(HQm+n−3)P = (PHQm−2)(PQn−1),  
L(Qm+n−2)L = (LQm−1)(LQn−1).  
If we equate the corresponding entries on both sides of the matrix equations,  
we obtain Equations (26) and (27) respectively.  
Corollary 1. For a positive integer n, the following equality holds:  
ꢁ ꢂ ꢁ ꢂ  
2
2
(k)  
BCFn+1  
(k)  
2n+1  
(k)  
¯
¯
+ BCFn  
= (β1(k)e1 + β(k)e2)(β  
e1 + β(k) e2)  
1
2n+1  
+(β0(k)e1 + β(k)e2)(β(k)e1 + β(k)e2).  
¯
¯
0
2n  
2n  
       
On bicomplex Fibonacci finite operator sequences  
256  
Proof. Substituting m −→ n into the equation (27) and using equation (8)  
we get  
ꢁ ꢂ ꢁ ꢂ  
2
2
(k)  
BCFn+1  
(k)  
+ BCFn  
=BCF2(k)BCF2(nk) + BCF1(k)BCF2n−1  
(k)  
ꢁ ꢂ  
(k)  
= BCF1(k) + BCF0  
BCF2(nk) + BCF1(k)BCF2n−1  
(k)  
ꢁ ꢂ  
(k)  
(k)  
(k)  
=BCF1  
BCF2(nk) + BCF2n−1 + BCF0(k)BCF2n  
=BCF1(k)BCF2n+1 + BCF0(k)BCF2n  
(k)  
(k)  
=(β1(k)e1 + β(k)e2)(β  
e1 + β(k) e2)  
(k)  
2n+1  
¯
¯
1
2n+1  
+ (β0(k)e1 + β(k)e2)(β(k)e1 + β(k)e2)  
¯
¯
0
2n  
2n  
Corollary 2. For positive integer n, the following equality holds:  
ꢁ ꢂ ꢁ ꢂ  
2
2
(k)  
BCFn+1  
(k)  
− BCFn−1  
= (β1(k)e1 + β(k)e2)(β(k)e1 + β(k)e2)  
¯
¯
1
2n  
2n  
+ (β0(k)e1 + β(k)e2)(β  
e1 + β(k) e2).  
(k)  
2n−1  
¯
¯
0
2n−1  
Proof. Firstly, we get  
ꢃ
ꢄ
ꢁ ꢂ ꢁ ꢂ ꢁ ꢂ ꢁ ꢂ  
2
2
2
2
(k)  
BCFn+1  
(k)  
− BCFn−1  
(k)  
BCFn+1  
(k)  
=
+ BCFn  
ꢃ
ꢄ
ꢁ ꢂ ꢁ ꢂ  
2
2
(k)  
(k)  
−
BCFn  
+ BCFn−1  
.
(28)  
Next, we make the following computations.  
" #  
(k)  
h i  
ꢁ ꢂ ꢁ ꢂ  
BCFn+1  
2
2
(k)  
BCFn+1  
(k)  
(k)  
(k)  
+ BCFn  
=
=
=
BCFn+1 BCFn  
(k)  
BCFn  
" #  
(k)  
h i  
BCF1  
(k)  
(k)  
QnQn  
BCF1  
BCF0  
(k)  
BCF0  
" #  
(k)  
h i  
BCF1  
(k)  
(k)  
Q2n  
.
(29)  
BCF1  
BCF0  
(k)  
BCF0  
 
I. Das, B.C. Tripathy, M. Sen  
257  
Similarly, we also have  
" #  
(k)  
h i  
ꢁ ꢂ ꢁ ꢂ  
BCFn  
2
2
(k)  
+ BCFn−1  
(k)  
(k)  
(k)  
BCFn−1  
BCFn  
=
=
=
BCFn  
(k)  
BCFn−1  
" #  
(k)  
h i  
BCF1  
(k)  
(k)  
Qn−1Qn−1  
BCF1  
BCF0  
(k)  
BCF0  
" #  
(k)  
h i  
BCF1  
(k)  
(k)  
Q2n−2  
.
(30)  
BCF1  
BCF0  
(k)  
BCF0  
By using equations(28)-(30), we have  
ꢃ
ꢄ
ꢁ ꢂ ꢁ ꢂ ꢁ ꢂ ꢁ ꢂ  
2
2
2
2
(k)  
BCFn+1  
(k)  
− BCFn−1  
(k)  
BCFn+1  
(k)  
=
+ BCFn  
ꢃ
ꢄ
ꢁ ꢂ ꢁ ꢂ  
2
2
(k)  
(k)  
−
BCFn  
+ BCFn−1  
" #  
(k)  
h i  
BCF1  
(k)  
(k)  
=
Q2n  
BCF1  
BCF0  
(k)  
BCF0  
" #  
(k)  
h i  
BCF1  
(k)  
(k)  
−
Q2n−2  
BCF1  
BCF0  
(k)  
BCF0  
" #  
(k)  
h i  
ꢁ ꢂ  
BCF1  
(k)  
(k)  
=
=
Q2n − Q2n−2  
BCF1  
BCF0  
(k)  
BCF0  
" #  
(k)  
h i  
ꢁ ꢂ  
BCF1  
(k)  
(k)  
Q2n−2 Q2 − I  
.
BCF1  
BCF0  
(k)  
BCF0  
By Cayley-Hamilton theorem it follows that,  
Q2 − Q − I = [0]2×2  
.
 
On bicomplex Fibonacci finite operator sequences  
258  
Also we get  
"
#
"
(k)  
BCF1  
h
h
h
h
i
i
i
i
ꢄ
ꢅ
ꢄ
ꢅ
2
2
(k)  
(k)  
(k)  
BCF1  
(k)  
BCF0  
BCFn+1 − BCFn−1  
=
=
=
=
Q2n−2  
Q
(k)  
BCF0  
"
#
ꢇ
(k)  
BCF1  
Q2n−1  
(k)  
BCF1  
(k)  
BCF0  
(k)  
BCF0  
#
ꢆ
(k)  
F2n  
F2n−1  
F2n−1  
BCF1  
(k)  
BCF1  
(k)  
BCF0  
(k)  
BCF0  
F2n  
"
#
(k)  
BCF2n  
(k)  
BCF1  
(k)  
BCF0  
(k)  
BCF2n−1  
(k)  
=BCF1(k)BCF2(nk) + BCF0(k)BCF2n−1  
=(β1(k)e1 + β(k)e2)(β2(kn)e1 + β2(kn)e2) + (β(k)e1 + β(k)e2)  
¯
¯
¯
0
1
0
(β2n−1e1 + β2(kn)−1e2).  
(k)  
¯
Theorem 9. For the bicomplex Fibonacci finite operator, we have  
ꢄ
ꢅꢄ  
(k)  
(k)  
(k)  
(k)  
(k) (k)  
BCFn+rBCFn+s − BCFn(k)BCFn+r+s =(−1)nFr (F1 )2 − F0 F2  
(β1e1 + β1e2)  
¯
ꢅ
¯
¯
¯
(βse1 + βse2) − (β0e1 + β0e2)(βs+1e1 + βs+1e2) .  
Proof. With the use of equations (24) and (25), we can compute :  
ꢇ ꢈ  
h i h i  
Fr  
0
1
(k)  
(k)  
(k)  
(k)  
=
BCFn+r BCFn  
BCFn+1 BCFn  
Fr−1  
" #  
ꢇ ꢈ  
(k)  
(k)  
ꢊ ꢋ  
Fn+1  
Fn  
Fr  
0
1
= BCF1 BCF0  
,
(k)  
(k)  
Fn−1  
Fr−1  
Fn  
and  
" # " #  
ꢇ ꢈ  
(k)  
BCFn+s  
(k)  
BCFn+s  
1
0
=
=
(k)  
−BCFn+r+s  
−Fr−1 Fr  
−BCFn+s+1  
" #  
ꢇ ꢈ ꢇ ꢈ  
(k)  
Fn−1  
(k)  
1
0
−Fn  
BCFs  
.
(k)  
(k)  
Fn+1  
−Fr−1 Fr  
−BCFs+1  
−Fn  
I. Das, B.C. Tripathy, M. Sen  
259  
Also, we have the following computation:  
" #  
" #  
ꢇ ꢈ ꢇ ꢈ  
(k)  
Fn+1  
(k)  
(k)  
Fn−1  
(k)  
Fn  
Fr  
0
1
1
0
−Fn  
(k)  
(k)  
Fn−1  
(k)  
(k)  
Fn+1  
Fr−1  
−Fr−1 Fr  
Fn  
−Fn  
" # " #  
(k)  
(k)  
(k)  
Fn−1  
(k)  
Fn+1  
Fn  
−Fn  
=
(FrI)  
(k)  
(k)  
Fn−1  
(k)  
(k)  
Fn+1  
Fn  
−Fn  
" # " #  
(k)  
(k)  
(k)  
(k)  
Fn+1  
Fn  
Fn−1  
−Fn  
= Fr  
= Fr  
(k)  
(k)  
Fn−1  
Fn+1Fn−1 − (Fn  
(k)  
(k)  
Fn+1  
Fn  
−Fn  
"
#
(k)  
(k)  
(k)  
2
)
0
(k)  
(k)  
(k)  
0
−(Fn )2 + Fn−1Fn+1  
ꢁ
ꢂ
(k)  
(k)  
= −Fr (F(k))2 − Fn−1Fn+1  
I
n
ꢁ ꢂ  
(k) (k)  
= (−1)nFr (F(k))2 − F0 F2  
I.  
1
Then, we get  
(k)  
(k)  
(k)  
BCFn+rBCFn+s − BCFn(k)BCFn+r+s  
" #  
(k)  
h i  
BCFn+s  
(k)  
(k)  
=
BCFn+r BCFn  
(k)  
−BCFn+r+s  
ꢇ ꢈ  
ꢁ ꢂ ꢊ ꢋ  
BCFs  
= (−1)nFr (F(k))2 − F0 F2  
BCF1 BCF0  
(k) (k)  
1
−BCFs+1  
ꢁ ꢂ  
= (−1)nFr (F(k))2 − F0 F2  
(BCF1BCFs − BCF0BCFs+1  
(β1e1 + β1e2)(βse1 + βse2)  
)
(k) (k)  
1
ꢁ ꢂꢁ  
= (−1)nFr (F(k))2 − F0 F2  
(k) (k)  
¯
¯
1
ꢂ
¯
¯
− (β0e1 + β0e2)(βs+1e1 + βs+1e2) .  
Corollary 3. Equations (22) and (23) are obtained by substituting n by n−1  
and r = s = 1 in the aforementioned theorem.  
3 Conclusions  
In this paper, we have presented a systematic study of bicomplex Fibonacci  
finite operator sequences through the idempotent representation of bicom-  
plex numbers. Unlike previous works on hypercomplex Fibonacci-type struc-  
tures that mainly relied on classical component-wise formulations, our ap-  
proach is based explicitly on the orthogonal idempotent decomposition in-  
trinsic to the bicomplex algebra. The main novelty of this work lies in  
On bicomplex Fibonacci finite operator sequences  
260  
deriving recurrence relations, polynomial representations, and matrix repre-  
sentations directly in terms of the idempotent basis. This framework simpli-  
fies the structural analysis by decomposing the sequences into independent  
complex components, leading to clearer proofs and new structural identities  
that are not immediately visible in the standard representation. It is worth  
noting that, unlike quaternion algebra introduced by William Rowan Hamil-  
ton —which is noncommutative and does not admit a non-trivial idempo-  
tent representation— the bicomplex algebra is commutative and possesses  
orthogonal idempotent elements. This distinctive feature makes the idem-  
potent method particularly effective for studying operator sequences. The  
results obtained here provide a refined structural perspective and open new  
directions for further investigations on recursive operator sequences within  
the bicomplex framework and related commutative hypercomplex systems.  
Acknowledgments. Dedicated to the loving memory of my mother,  
Nirupama Das, and my maternal grandparents, Bholanath Das and Basanti  
Rani Das.  
The first author gratefully acknowledges the Ministry of Education, Gov-  
ernment of India, for providing financial support through the Institute Fel-  
lowship (GATE). The authors also sincerely thank the reviewers for their  
valuable comments and suggestions, which have significantly improved the  
presentation of the paper.  
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