Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 
BİTLİS EREN UNIVERSITY JOURNAL OF SCIENCE 
ISSN: 2147-3129/e-ISSN: 2147-3188 
VOLUME: 11 NO: 2 PAGE: 586-593 YEAR: 2022 
DOI: 10.17798/bitlisfen.1063550 
Algebraic Construction for Dual Quaternions with  
 
Gülsüm Yeliz ŞENTÜRK1*, Nurten GÜRSES2, Salim YÜCE2 
 
1Department of Computer Engineering, Faculty of Engineering and Architecture,  
 Istanbul Gelisim University, 34310, Istanbul, Turkey 
2Department of Mathematics, Faculty of Arts and Sciences,  
 Yildiz Technical University, 34220, Istanbul, Turkey 
(ORCID: 0000-0002-8647-1801) (ORCID: 0000-0001-8407-854X) (ORCID: 0000-0002-8296-6495) 
 
 
Keywords:  Abstract 
Generalized complex number, In this paper, we explain how dual quaternion theory can be extended to dual 
Dual quaternion, Matrix form. quaternions with generalized complex number    components. More 
specifically, we algebraically examine this new type dual quaternion and give several 
matrix representations both as a dual quaternion and as a .  
 
 
 
1. Introduction form with different multiplication conditions for 
 quaternionic units as [4-10]: 
A real quaternion, as an extension of complex number  
in four dimensions, is defined as           2i  2j  2k  0,  
 (1) 
 ij  ji  jk  kj  ki  ik  0.
q  a0 a1e1 a2e2 a  3e3,   
 The set of dual quaternions , which is isomorphic 
where a ,a ,a ,a  are real components and  0 1 2 3 e1,e2,e3 to Galilean 4-space, forms a commutative division 
are non-real quaternionic units with the following algebra under addition and multiplication [7]. 
multiplication schema [1-3]: Furthermore, using the dual quaternions, one can 
 express the Galilean transformation in one 
e 2  e 2  e 2  1,  quaternionic equation. 1 2 3
From a different viewpoint, the set of 
e1e2  e2e1  e3 ,  
 generalized complex numbers ( ), the general  
e e  e e  e ,  bidimensional hypercomplex system, is denoted by 2 3 3 2 1
e e  e e  e . q,p  and defined by the ring [11-16]: 3 1 1 3 2
  
The set of real quaternions, which is isomorphic to [X ] z  x1  x2I :I
2  Iq p, I  ,
 ,  
Euclidean 4-space, forms a non-commutative and an 2    X qX  p             p,q, x1, x2  
associative algebra under addition and multiplication. 
The real quaternions have many applications such as  
describing rotations in robotics and computer where I  is the generalized complex unit. It is 
animation with rotation axis and angle. isomorphic (as ring) to the following types 
2
A dual quaternion, as an extension of dual considering the sign of   q  4p : for   0  
number in four dimensions, is defined by the same hyperbolic system, for   0  elliptic system and for 
                                                          
*Corresponding author: gysenturk@gelisim.edu.tr                           Received: 26.01.2022, Accepted: 26.06.2022 
586 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
  0  parabolic system. The canonical forms of Here, I  commutes with the three dual 
these systems are given by, respectively, quaternion units. One can see that, the usual dual 
 hyperbolic (perplex, split complex, double) operator distinct from the dual quaternion units for 
numbers q  0,p 0 . 0,1  [17-20], 
Throughout the paper, 
 complex (ordinary) numbers 0,1  [20, 21], q  a0 a1ia ja ,  2 3k p  b0 b1ib2jb3k
 dual numbers 0,0  [20, 22, 23]. and r  c0  c1i  c2j c3k  are taken. 
Specially, dual numbers have been widely used for 
We firstly define the basic algebraic 
the search of closed form solutions in the fields of 
displacement analysis, kinematic synthesis, and operations on dual quaternions with  
dynamic analysis of spatial mechanisms. components. For any q , S  a  is the scalar q 0
In q,p , the value part and V  a i  a j a k  is the vector part. q 1 2 3
z  zz  (x1  x2I )(x1  x2I )  x
2
1  px
2
2 qx1x2  Equality is as follows: p  q  S p  Sq ,  Vp Vq. 
is referred to as the characteristic determinant of z . 
The addition of q  and p  is defined as: 
Considering this characteristic value, z  is called 
 
timelike for  0 , spacelike for  0  and null z z q  p  a0 b0  a1 b1  i
for  0  [12].  z
Number systems play a special role in defining            a2 b2  j a3 b3 k   
different types of quaternions. Combining          S p  Sq Vp Vq
fundamental properties of numbers and quaternions 
enables to determine new features. Considering the          S pq Vpq .
numbers mentioned above, the quaternions with 
The quaternion q  a a i a ja
different number components have been studied by 0 1 2 3
k  Sq Vq  is 
several authors in many points of view [24-30]. One called the conjugate of q . Furthermore, for c q,p,  
can see the combination of the dual numbers and the the scalar multiplication of c  and q  is defined as: 
real quaternions in the studies [27, 28, 30]. Moreover, 
 
as an application, the representational method based 
on quaternions with dual number coefficients related cq  ca0  ca1i  ca2j ca3k
  
to electromagnetism can be seen in [31, 32].      cSq  cVq .
In this paper, we are interested in the combination 
 
of dual quaternions and . In Section 2, we 
The multiplication of q  and p  is defined as: 
extend definitions and some universal known results 
 
of dual quaternions to dual quaternions with  
components. Finally, we provide a complete  q p  a0b0  a0b1  a1b0  i
classification in conclusion.         a0b2  a2b0  j a0b3  a3b0 k
  (2) 
2. Dual Quaternions with  Components      SqS p  SqVp  S pVq
      pq.
This original section discusses an algebraic behavior 
 
of dual quaternions with components. Also it 2
Additionally, N  qq  qq  a  is called the norm 
proceeds with the examination of several matrix q 0
representations. 
q 1
q
 of . Hence, the quaternion (q)   is called the 
N
Definition 1. The dual quaternion with  q
components is of the form: inverse of q  for non-null Nq  that is N  0 . As it q
q  a0 a1ia2ja3k,   is seen many properties of dual quaternions with 
  components are familiar with the usual dual 
where the dual quaternion units satisfy equations in quaternions.  
(1). The set of these quaternions is denoted by .   
587 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
Standard elementary conjugate properties establish More specially, we examine some identities for the 
the following proposition. scalar product. 
  
Proposition 1. For any q, p  and c ,c  , Proposition 3. For any q, p  and r  , the 1 2
the followings hold:  followings hold: 
q  q i)  , qr, pr  rq,rp  rq, pr  qr,rp , i)
             N q, p,  
ii) c1 p  c2q  c1 p  c2 q , 
r
ii) rq, p  q,r p  q,rp . 
iii) q p  pq  q p , 
 
2
iv) Nc q  c1 Nq , Proof: Considering the scalar product and the norm, 1
v) N  N N  N N . the following proofs can be conducted:  q p q p p q
 i) qr, pr  a
2
0c0 b0c0   c0 a0b0   
Proof: Considering the conjugate, properties i) and ii)              Nr q, p,  
are quickly obvious.  
 ii) qr, p  a0c0 b0  a0 b0c0   q, pr.  
iii) By using equation (2), we have:   
 We are now ready to prove the results based on matrix 
qp  a b  a b  a b  i approach. 0 0 0 1 1 0
   
        a0b2  a2b0  j a0b3  a3b0 k. Theorem 1. Every element q  of  can be 
 
represented by a quaternionic matrix: 
So, it is verified that q p  pq  q p .  
 iv) By having property ii) and the norm, we have: a0  a3k a1i  a2j
E  .
Nc  c q
 (3) 
c q  c2N . q  
1q 1 1 1 q a1i  a2j a0  a3k
 v) From property iii), we get:  
Nq p  q pq p  q p pq  NqN p  N p Nq .  Hence  2 ( ) .  
  
Proposition 2.  is a 4 -dimensional module over Proof: For q , :  , q Eq  is a linear 
q,p  and an 8 -dimensional vector space over  map, where 
 
with bases 1, i, j,k  and 1, I , i, Ii, j, Ij,k, Ik , 
 a0  a3k a1i  a j2
respectively.  : E  q  2 ( ):  Eq     
  a1i  a2j a0  a3k
Definition 2. For any q, p , the scalar product is a subset of 2 ( ) . So, one can realize the 
is given by: correspondence between  and  by the map . 
 
So it is no surprised that 22  representation of q  is 
  q,p
  Eq . The proof is completed. 
   q, p q, p  SqS p  a0b0  S pq  
 Corollary 1. For all q , (q)  can also be 
and the vector product is defined by: 
 written as follows: 
 
 
  (q)  a0I2 a1Ia  2Ja3K,  
   q, p q p  SqV  S Vq V .p p q p  
 where (i)  I,  (j)  J,  (k) K  satisfy 
equations in (1). 
588 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
Theorem 2. For q, p  and  , then the  a0b0 0 0 0 
followings hold:  a0b 1
 a1b0 a0b0 0 0 
i) q  p E E , qp  .   q p a b  a b 0 a b 0 0 2 2 0 0 0
ii) E ,  q p Eq Ep a0b3  a3b0 0 0 a0b0 
iii) Eq   Eq  ,  
Also, we have: 
iv) E . qp EqEp  
Proof: iv) For q, p , using equations (2) and (3), a0 0 0 0  b0 0 0 0 
we can write:    a
 1
a0 0 0 b1 b0 0 0  
 q p  
 a2 0 a0 0
 b2 0 b 0  a0b0  a0b3  a3b0 k a b  a b  i  a  00 1 1 0 0b2  a2b0  j
Eqp        
a0b1  a1b0  i  a0b2  a2b0  j a0b0  a0b3  a3b0 k  a3 0 0 a0  b3 0 0 b0 
Moreover, we obtain: 
 a0b0 0 0 0    
 
a0  a3k a1i  a2j b0 b3k b i b a b  a b1 2j
E E   0 1 1 0
a0b0 0 0
q p     
a1i  a2j a0  a3k b1i b2j b0 b3k
          

 a 0b2  a2b0 0 a0b0 0
 a b  a b  a b k a b  a b  i  a b 0 0 0 3 3 0 0 1 1 0 0 2  a2b0  j  
   .
a0b1  a1b0  i  a 
a0b3  a3b0 0 0 a0b0 
0b2  a2b0  j a0b0  a0b3  a3b0 k 
         p q .
It is clear that E E E . The other properties can qp q p  
be proved similarly. It is obvious that qp  p q  q p . 
Theorem 3. Every element q  of  can be  
represented by the following matrix: v) Considering equation (4) and 
   diag(1,1,1,1) , we get: 
a0 0 0 0 
 
  1 0 0 0  b0 0 0 0  1 0 0 0 
a a 0 0    
 1 0  0 1 0 0 b b 0 0
 
0 1 0 0
 .  (4)       1 0   q p ,a2 0 a0 0 
0 0 1 0  b2 0 b0 0  0 0 1 0 
 
     
  0 0 0 1 b3 0 0 b0  0 0 0 1
a3 0 0 a0    b0 0 0 0 
  b1 b0 0 0
           .
So,  is subset of 4 ( q,p) . Moreover, for 
b 0 b 0 2 0
 
b3 0 0 b0 
q, p  and  ,   
i) p  q  p  q , One can see that the final matrix is the matrix , so p
ii) pq  p  q , we can write  p  . p
iii)  p  ( p ) , The proofs of the other properties are straightforward 
iv)   by considering 44  real matrix representation of the pq p q q p , 
dual quaternions. 
v)  p   where   diag(1,1,1,1) , p  
2 Definition 3. The column matrix form of p  with 
and det( p )  N  p . 
T
respect to {1, i, j,k}  is p  b b b b  .   
Proof: iv) For q, p , using equations (2) and (4), 0 1 2 3
 
we obtain: 
Corollary 2. Using the above definition, the 
 
multiplication of q  and p  is also calculated as: 
qp  q p   pq .  
 
589 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
Corollary 3. For q ,  
fq 1  q  x01  x02I  x11i  x12Ii  x21j
q  a0I4  a1 a2  a  3K,  x22Ij x31k  x32Ik,
where 
fq  I   qI  px02  (x01  qx02 )I  px12i  x11  qx12  Ii  px22 j
 
 x21  qx22  Ij px32k  x31  qx32  Ik,
0 0 0 0 0 0 0 0 0 0 0 0
 

f
      q
i  qi  x01i  x02Ii,  
1 0 0 0 0 0 0 0 0 0 0 0  
    ,    ,K    . fq  Ii  qIi  px02i  (x 01
 qx02 )Ii,
0 0 0 0 1 0 0 0 0 0 0 0 fq  j  qj  x01j x02Ij,  
     
0 0 0 0 0 0 0 0 1 0 0 0 fq  Ij  qIj  px02 j (x01  qx02 )Ij,
The following theorem indicates how to calculate the fq k   qk  x01k  x02Ik,
formula for matrix representation of the inverse of fq  Ik   qIk  px02k  (x01  qx02 )Ik.
q .  
Hence, by concerning the standard basis 
1
Theorem 4. Let q  and q  be the inverse of q.  
1, I , i, Ii, j, Ij,k, Ik , we have 88  real matrix 
1
Then, 1   for non-null detq q  q   representation of q  is calculated as in equation 
det( q )
(5). The proof of the properties can be conducted by 
that is  0 .  
det q  considering the above linear map. Specially, for 
property iv), by taking  
Theorem 5. According to 1, I , i, Ii, j, Ij,k, Ik , the 
real matrix representation of ai  xi1  xi2I ,  b  y , 0  i  3  i i1  yi2I  q,p  
q  a0 a1ia2ja  is: 3k for q, p  and using equations (2) and (5), the 
 
x px 0 0 0 0 0 0  multiplication of q  and p  gives the matrix  01 02 qp
 
x
 02
x01 qx02 0 0 0 0 0 0   
 x11 px 
quickly.  
12 x01 px02 0 0 0 0
 
x12 x11 qx12 x02 x01 qx02 0 0 0 0
q 
 ,
x21 px22 0 0 x01 px02 0 0
 (5) With an alternative thought, 
 
x22 x21 qx22 0 0 x02 x01 qx02 0 0 
 q  a a ia ja k , a  x  x I  , can x31 px 0 0 0 0 x px  0 1 2 3 i i1 i2 q,p 32 01 02 
x32 x31 qx32 0 0 0 0 x02 x01 qx02 
be written as q  q  q I  in  where 
 0 1
where a  x  x I  q  x  x i  x j x k  for 0  i  3 , i i1 i2 q,p , 0  i  3 . Moreover, j1 0 j 1 j 2 j 3 j
for q, p  and  ,  1 j  2 . So,  is a 2 -dimensional module over 
i) p  q   ,  with base 1, I . This consideration provides a p q
ii)   , reformulation of the previous results. pq p q
 
iii)  p  ( p ) , Theorem 6. Let q  q0  q1I , p  p0  p1I   
iv) qp  q p . 
and  . Every element of  is written by a 
Proof: Let us define the linear map fq  from  to 22  dual quaternion matrix: 
subset of ( )  such that fq  p  qp  for every 
 
8
q0 pq1 
  
p . By taking a  x  x I  , 0  i  3 , q  q 
.  
i i1 i2 q,p q1 q0  q1
we have the following equations:  It means that  is subset of ( ) . So, we have: 2
i) p  q  p  q , 
ii) pq  p  q , 
iii)  p  ( p ) , 
iv) pq  p q , 
590 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
and det( )  q2 qq q  pq2 , where det  Using the different types of quaternions are the way q 0 0 1 1
to description of the classical and quantum fields and 
corresponds the determinant of the quaternion 
reasonable to express space-time transformations. In 
matrix2. Moreover,  q I  q I,  where q 0 2 1 terms of the hyperbolic quaternion, the general 
0 p Lorentz space-time transformation can be discussed. 
I    .  (It is worth noting that there exists With similar thought, the dual quaternions can be 
1 q expressed for discussing the Galilean transformation. 
q 1 In terms of the dual quaternions this transformation 
different ways to take I , for instance: I    , see p 0 with underlying algebraic features enables an   efficient form [7-9].  
in [34]). With the leading of the above discussions, 
 
considering  as components of dual 
Definition 4. The column matrix form of p  with 
quaternions is the main motivation of this study. For 
T
respect to 1, I  is p   p p  . this purpose, we construct dual quaternions with 0 1
  coefficients for real p,q . Moreover, we 
Corollary 4. By using above definition, we obtain examine the basic structures and algebraic properties 
q p  p   pq.   by writing them in two forms: a  and a q
quaternion. Additionally, we established 22 , 44  
 
Definition 5. The matrix and 88  matrix representations. 
 With this approach, we can easily write the dual 
quaternions with elliptic, parabolic and hyperbolic 
T
T T q0 q  q  q     number components considering   0,   0  and  0 1    81( )  
q1  2  0 , respectively, where  q 4p.  Bearing in 
 is called as the vector form of q , where mind the special values p  and q , we have several 
 
types of dual quaternions with  components. 
q  x  x i  x j x k  j1 0i 1 j 2 j 3 j  For q  0,  p 1 dual quaternions with complex 
T
T
and q  x , x , x , x   x x x x  number components, for q  p  0  dual quaternions j1 0 j 1 j 2 j 3 j  0 j 1 j 2 j 3 j 
are vectors (matrices) for 1 j  2 . with dual number components and for q  0, p1 
 dual quaternions with hyperbolic number components 
4. Conclusion are obtained.  
  
Quaternions ([1-3]) have a deep mathematical Contributions of the authors 
meaning with a long history dating back and are used  
in physics to clarify the formulation of physical laws. Every author contributed equally to this work. 
A milestone moment in the use of quaternions in  
theoretical physic is the creation of special relativity, Conflict of Interest Statement 
which unifies space and time to form a 4-dimensional  
space-time. By replacing real quaternions with There is no conflict of interest between the authors. 
complex ones offers a valuable tool in creating  
classical physical laws. Complex quaternions having Statement of Research and Publication Ethics 
several properties allow the desirable theorems of  
modern algebra to be applied. Furthermore, an The author declares that this study complies with 
important extension of real quaternions are the Research and Publication Ethics. 
hyperbolic quaternions and the dual quaternions. 
 
                                                          
2  a b  

det  , [35].    dacb
 c d

 
591 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
References 
 
[1] W. R. Hamilton, Elements of quaternions. New York: Chelsea Pub. Com.,1969. 
[2] W. R. Hamilton, Lectures on quaternions. Dublin: Hodges and Smith, 1853. 
[3] W. R. Hamilton, “On quaternions; or on a new system of imaginaries in algebra,” The London, 
Edinburgh and Dublin Philosophical Magazine and Journal of Science (3rd Series), xxv-xxxvi, 1844–
1850. 
[4] W. K. Clifford, “Preliminary sketch of bi-quaternions,” Proceedings of the London Mathematical 
Society, vol. s1–4, no. 1, pp. 381–395, 1873. 
[5] J. D. Jr. Edmonds, Relativistic reality: A modern view. Singapore: World Scientific, 1997. 
[6] Z. Ercan and S. Yüce, “On properties of the dual quaternions,” European Journal of Pure and Applied 
Mathematics, vol. 4, no. 2, pp. 142–146, 2011. 
[7] V. Majernik, “Quaternion formulation of the Galilean space-time transformation,” Acta Physica 
Slovaca, vol. 56, pp. 9–14, 2006. 
[8] V. Majernik and M. Nagy, “Quaternionic form of Maxwell’s equations with sources,” Lettere al Nuovo 
Cimento, vol. 16, pp. 165–169, 1976. 
[9] V. Majernik, “Galilean transformation expressed by the dual four-component numbers,” Acta Physica 
Polonica, vol. 87, no. 6, pp. 919–923, 1995. 
[10] Y. Yaylı and E. E. Tutuncu, “Generalized Galilean transformations and dual quaternions,” Scientia 
Magna, vol. 5, no. 1, pp. 94–100, 2009. 
[11] I. Kantor and A. Solodovnikov, Hypercomplex numbers. New York: Springer-Verlag, 1989. 
[12] F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti and P. Zampetti, The mathematics of 
Minkowski space-time and an introduction to commutative hypercomplex numbers. Birkhäuser Basel, 
2008. 
[13] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, “Two-dimensional hypercomplex numbers and related 
trigonometries and geometries,” Advances in Applied Clifford Algebras, vol. 14, pp. 47–68, 2004. 
[14] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, “N-dimensional geometries generated by 
hypercomplex numbers,” Advances in Applied Clifford Algebras, vol. 15, no. 1, 1–25, 2005. 
[15] A. A. Harkin and J. B. Harkin, “Geometry of generalized complex numbers,” Mathematics Magazine, 
vol. 77, no. 2, pp. 118–129, 2004. 
[16] S. Veldsman, “Generalized complex numbers over near-fields,” Quaestiones Mathematicae, vol. 42, no. 
2, pp.181–200, 2019. 
[17] W. K. Clifford, Mathematical papers, R. Tucker, Ed., New York: Chelsea Pub. Co., Bronx, 1968. 
[18] P. Fjelstad, “Extending special relativity via the perplex numbers,” American Journal of Physics, vol. 
54, no. 5, pp. 416–422, 1986. 
[19] G. Sobczyk, “The hyperbolic number plane,” The College Mathematics Journal, vol. 26, no. 4, pp. 268–
280, 1995. 
[20] I. M. Yaglom, A simple non-Euclidean geometry and its physical basis. New York: Springer-Verlag, 
1979. 
[21] I. M. Yaglom, Complex numbers in geometry. New York: Academic Press, 1968. 
[22] E. Pennestrì and R. Stefanelli, “Linear algebra and numerical algorithms using dual numbers,” 
Multibody System Dynamics, vol. 18, no. 3, pp. 323–344, 2007. 
[23] E. Study, Geometrie der dynamen. Leibzig: Mathematiker Deutschland Publisher, 1903.  
[24] W.R. Hamilton, “On the geometrical interpretation of some results obtained by calculation with 
biquaternions”, In Proceedings of the Royal Irish Academy, vol. 5, pp. 388–390, 1853. 
[25] Y. Tian, “Biquaternions and their complex matrix representations,” Beiträge zur Algebra und 
Geometrie/Contributions to Algebra and Geometry, vol. 54, no. 2, pp. 575–592, 2013. 
[26] E. Karaca, F. Yılmaz and M. Çalışkan, “A unified approach: split quaternions with quaternion 
coefficients and quaternions with dual coefficients,” Mathematics, vol. 8, no. 12, 2149, 2020. 
[27] A. P. Kotelnikov, Screw calculus and some applications to geometry and mechanics. Kazan: Annal. 
Imp. Univ., 1895. 
[28] A. McAulay, Octonions: a development of Clifford's biquaternions. University Press, 1898. 
[29] M. Jafari, “On the properties of quasi-quaternion algebra,” Communications Faculty of Sciences 
University of Ankara Series A1 Mathematics and Statistics, vol. 63, no. 1, pp. 1-10, 2014. 
592 
 
G. Y. Şentürk, N. Gürses, S. Yüce / BEÜ Fen Bilimleri Dergisi 11 (2), 586-593, 2022 
 
[30] L. Qi, C. Ling and H. Yan, “Dual quaternions and dual quaternion vectors,” Communications on Applied 
Mathematics and Computation, pp. 1–15, 2022 
[31] S. Demir and K. Özdas, “Dual quaternionic reformulation of electromagnetism,” Acta  Physica 
Slovaca, vol. 53, no. 6, pp. 429–436, 2003. 
[32] S. Demir, “Matrix realization of dual quaternionic electromagnetism,” Central European Journal of 
Physics, vol. 5, no. 4, pp. 487–506, 2007. 
[33] F. Zhang, “Quaternions and matrices of quaternions,” Linear algebra and its Applications, vol. 251, pp. 
21–57, 1997. 
[34] F. Messelmi, “Generalized numbers and their holomorphic functions,” International Journal of Open 
Problems in Complex Analysis, vol. 7, no. 1, pp. 35–47, 2015. 
 
593