EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6492 ISSN 1307-5543 – ejpam.com Published by New York Business Global Product Difference Fibonacci Identities Revisited: Quaternionic Generalizations of Everman and Koshy Bahar Demirtürk1,∗, Nazim Topal2 1 Department of Fundamental Sciences, Faculty of Engineering and Architecture, Izmir Bakırçay University, Izmir, Türkiye Abstract. In this paper, we investigate new identities involving generalized Fibonacci and Lucas sequences, as well as their associated quaternions. After establishing the fundamental properties of these generalized number sequences, we derive quaternionic extensions of product-difference identities originally introduced by Everman and Koshy for Fibonacci numbers. These results not only generalize classical identities, but also reveal new algebraic structures within the framework of generalized quaternion sequences. 2020 Mathematics Subject Classifications: 15B33, 11B39, 11B75, 11Z05 Key Words and Phrases: Generalized Fibonacci sequences, generalized Fibonacci quaternions, Quaternions, product differences 1. Introduction Fibonacci and Lucas number sequences are used in many areas of mathematics due to their repetitive structures. They arise in number theory with their divisibility properties, in analysis with their generator functions, in linear algebra with their matrix representa- tions, in combinatorics with tiling problems, and in geometry with their fractal structures, etc. Because of such a wide range of applications, many generalized versions of these num- ber sequences have been defined; some by changing the recurrence relation, and others by changing the initial conditions. These sequences offer more complex and richer mathe- matical structures, especially when combined with quaternion structures. Furthermore, the extension of these numbers to quaternions is being studied by many authors in both theoretical and applied fields. Fibonacci sequences are seen in many structures in nature, such as the arrangement of sunflower seeds, the structure of a pine cone, the crystallization of a snowflake, and the spiral pattern in seashells. In [1], Sharma explores the evolution of the Fibonacci sequence and its modern applications, especially in fractal geometry. Generalized sequences can be ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6492 Email addresses: bahar.demirturk@bakircay.edu.tr (B. Demirtürk), nazimtopal@gmail.com (N. Topal) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 2 of 22 used to explain more complex models of these natural patterns, for example the structure of DNA [2]. In addition to being used to model patterns in quantum physics and other physical systems, quarks, which we know from physics theory, can also be given as an example[3]. Fibonacci and Lucas number sequences are notable for their mathematical structure as well as their applications in various disciplines. The properties of these number sequences are utilized in computer science, algorithm development, cryptography, nature science, art, engineering and design, finance and economics [4]. Among the algorithms used in com- puter science, the Fibonacci search algorithm and the golden ratio search algorithm offer particularly good solutions for searching and sorting. The increasing number of different generalizations of these sequences of numbers is being used to develop new algorithms [5]. Thus, more flexible and adaptable algorithm versions can be developed that reduce data processing costs, such as the running time of the algorithm used, and how close the results obtained are to the true value with less error. In addition, these sequences of numbers are used for key exchange and encryption in secure communication systems. Moreover, Fibonacci and Lucas sequences are used in engineering for signal processing and struc- tural optimization, as well as architecture and design [6]. The golden ratio is also used in determining aesthetic proportions in art and architecture, rhythmic structures and compo- sitions in music. In addition, Fibonacci sequences are used in technical analysis in finance and economics to determine support and resistance levels [7]. Generalized sequences can be used to build more complex models of market movements. There are many studies on the mathematical properties and applications of quater- nions whose elements are generalized Fibonacci and Lucas numbers. Research in this area not only provides new results in theoretical mathematics, but also new solutions in opti- mization, cryptography, and especially in transformation and rotation problems in physics and engineering. Halıcı obtained numerous new equations by generalizing various Fibonacci and Lucas quaternions and deriving their Binet formulas, generator functions, and matrix repre- sentations [8–10]. Kesim obtained exponential generator functions for the generalized Fibonacci and Lucas quaternions and obtained binomial sums of these quaternions [11]. Kome et al. introduced the modified generalized Fibonacci and Lucas quaternions and gave the generator functions, Binet formulas and matrix representations for these quater- nions [12]. Moreover, Aydinyüz and Asci introduced the generalized k-order Fibonacci and Lucas quaternions and obtained their generating functions and matrix representations [13]. Kızılateş et al. define higher-order generalized Fibonacci quaternions with q-integer components using q-integers and higher order generalized Fibonacci numbers and obtain Binet-like formulas, generator functions, recurrence relations, and matrix representations for these new quaternions in [14]. In [15], Shpakivskyi investigates some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions. In this paper, we rederive the classical results of Koshy and Everman using general- ized Fibonacci and Lucas numbers, and subsequently apply these results to quaternions. In this context, generalized Fibonacci and Lucas quaternions will be defined, and their properties will be studied in detail. In addition, generalized versions of some equations in B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 3 of 22 the literature will also be obtained in this study. The aim of this study is to understand the mathematical structures of quaternions based on generalized Fibonacci and Lucas numbers and to show their connection to classical results through the derivation of new identities. In this section some literature overwiev is given about Fibonacci and Lucas sequences, their generalizations, and quaternion generalizations. The rest of this paper is organized as follows. In the next section, the basic properties of generalized Fibonacci and Lucas sequences will be discussed and new identities related to these sequences will be obtained. Then in the third section, the algebraic properties of quaternions whose terms are com- posed of generalized Fibonacci and Lucas numbers will be given, and new identities re- lated to these quaternions will be obtained by using Binet formulas. Finally, we will give quaternion generalizations of product differences of Everman and Koshy based Fibonacci identities’. 2. Properties and Identities of Generalized Fibonacci and Lucas Sequences Definition 1. Let Hn be a sequence defined by the recurrence relation Hn = Hn−1 + Hn−2, for n ≥ 3, (1) with the initial conditions H1 = p, H2 = p + q, where p and q are arbitrary integers. The terms of this sequence are p, p + q, 2p + q, 3p + 2q, 5p + 3q, 8p + 5q, 13p + 8q, . . . . This sequence is called Horadam’s Generalized Fibonacci Sequence [16, 17]. (Fn) is the Fibonacci sequence with the recurrence relation F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2 for n ≥ 2, defined by taking p = 1 and q = 0 in (1). Similarly, taking p = 1 and q = 2 in (1), the Lucas sequence (Ln) is defined with the recurrence relation L0 = 2, L1 = 1, Ln = Ln−1 + Ln−2 for n ≥ 2, [18], [19]. Kalman and Mena, in [20], defined generalized Fibonacci and Lucas numbers with the recurrence relation An+2 = aAn+1 + bAn, for all n ≥ 0 where the sequences depend on initial conditions A0 and A1. They identified several number sequences corresponding to different values of R(a, b), as illustrated below. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 4 of 22 Table 1: Generalized Fibonacci and Lucas-Type Sequences [20] Sequence Initial Conditions First Terms Fibonacci numbers R(1, 1), A0 = 0, A1 = 1 {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . } Lucas numbers R(1, 1), A0 = 2, A1 = a {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, . . . } Pell numbers R(2, 1), A0 = 0, A1 = 1 {0, 1, 2, 5, 12, 29, 70, 169, . . . } Pell-Lucas numbers R(2, 1), A0 = 2, A1 = a {2, 2, 6, 14, 34, 82, . . . } Natural numbers R(2, −1), A0 = 0, A1 = 1 {0, 1, 2, 3, 4, 5, 6, . . . } Constant sequence R(2, −1), A0 = 2, A1 = a {2, 2, 2, 2, 2, 2, 2, . . . } Mersenne sequence R(3, −2), A0 = 0, A1 = 1 {0, 1, 3, 7, 15, 31, . . . } Fermat sequence R(3, −2), A0 = 2, A1 = a {2, 3, 5, 9, 16, 33, . . . } Periodic (with period=6) R(1, −1), A0 = 0, A1 = 1 {0, 1, 1, 0, −1, −1, 0, 1, . . . } Periodic (with period=6) R(1, −1), A0 = 2, A1 = a {2, 1, −1, −2, −1, 1, 2, 1, . . . } Even-indexed Fibonacci R(3, −1), A0 = 0, A1 = 1 {0, 1, 3, 8, 21, 55, . . . } Even-indexed Lucas R(3, −1), A0 = 2, A1 = a {2, 3, 7, 18, 47, . . . } Koshy, in [19], introduced the notion of (p, q) generalized Fibonacci numbers. In this study, we present several properties of the (p, q)-generalized Fibonacci and Lucas numbers. Now let us give the definition of a (p, q) generalization of Fibonacci and Lucas sequences. Definition 2. ([16, 17]) Let p, q ∈ Z. The sequence (Un) defined by the relation Un = pUn−1 + qUn−2, for all n ≥ 2 with initial conditions U0 = 0, U1 = 1 is called the generalized Fibonacci sequence, and the number Un is called the n − th generalized Fibonacci number. Definition 3. ([16, 17]) Let p, q ∈ Z.The sequence (Vn) defined by the relation Vn = pVn−1 + qVn−2, for all n ≥ 2 with initial conditions V0 = 2, V1 = p, is called the generalized Lucas sequence, and the number Vn is called the n − th generalized Lucas number. The characteristic equation of these generalized sequences are x2 − px − q = 0. Let ∆ = p2 + 4q > 0; the roots of this characteristic equation are given by α = p + √ ∆ 2 and β = p − √ ∆ 2 , which satisfy α + β = p and αβ = −q. Moreover, α2 = pα + q and β2 = pβ + q. By induction, it can be shown that for every n ∈ N, αn = αUn + qUn−1, (2) B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 5 of 22 βn = βUn + qUn−1. (3) Similarly, it can be shown by induction that for every n ∈ N, the elements of the generalized Lucas sequence satisfy the identities √ ∆ αn = αVn + qVn−1, (4) − √ ∆ βn = βVn + qVn−1. (5) Moreover, the Binet formulas for generalized Fibonacci and Lucas numbers are given by Un = αn − βn α − β , Vn = αn + βn (6) for all n ∈ Z [20, 21]. Since each term is a linear combination of the two preceding terms, we have U−1 = 1 q , U−2 = − p q2 , U−3 = p2 + q q3 , . . . and V−1 = −p q , V−2 = p2 + 2q q2 , V−3 = −p3 + 3pq q3 , . . . From these results, it was shown in [22], using the Binet formulas, that the generalized Fibonacci and Lucas numbers with negative indices satisfy U−n = −(−q)nUn, V−n = (−q)nVn. Many authors have used the properties of these sequences and have proved numerious identities, sum formulas, and matrix structures, while many others have investigated the product differences of Fibonacci and Lucas numbers. Some of these classical Fibonacci identities are such as Cassini’s identity F 2 n − Fn−1 Fn+1 = (−1)n−1, which was proved in 1680 and Catalan’s identity F 2 n − Fn−m Fn+m = (−1) n−m F 2 m, which was proved 1879, and the d’Ocagne’s identity Fm+n = Fm−1 Fn + Fm Fn+1, was proved in 1882. Since we will focus on the product differences of generalized Everman and Koshy based identities, let us first recall the term “Product Difference Fibonacci Identity” with the following definition. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 6 of 22 Definition 4. (Product Difference Fibonacci Identity) Let s ≥ 1, and the ai and bi be specified integers and Dn be of some interesting form for all integers n. Then the product of the form s∏ i=1 Fn+ai − s∏ i=1 Fn+bi = Dn(ai, bi; s) = Dn. (7) is introduced by Fairgrieve and Gould in [23]. It can be seen that Cassini’s and Catalan’s identities are some versions of the equation (7). There are many identities related to (7). For example, Morgado used the Catalan identity to prove the following equation Fn−2 Fn−1 Fn+1 Fn+2 − F 4 n = −1. in [24]. Recently, in [25], Melham discovered the following formula Fn+1Fn+2Fn+6 − F 3 n+3 = (−1)nFn. More generally the following equation Fn+a Fn+b − Fn Fn+a+b = (−1)n Fa Fb . (8) was stated by Everman at al. as a problem in The American Mathematical Monthly [26] and appears in Vajda [21, p. 177, Eq. (20a)]. (8) can be called the extended version of these classical Fibonacci identities. Actually, Horadam had already expressed some identities similar to equality (8) with some generalizations and he had stated the following equation Hn Hn+r+1 − H n−s H n+r+s+1 = (−1) n+s [p2 − p q − q2] F r+s+1 (9) in [16]. Then in [25] Melham proved the identity Fn+a+b−c Fn−a+c Fn−b+c − Fn−a−b+c Fn+a Fn+b = (−1) n+a+b+c Fa+b−c ( Fc Fn+a+b−c+(−1)c Fa−c Fb−c Ln ) . Since Cassini’s, Catalan’s, d’Ocagne’s identities, and all of the identities given in equa- tion (7), (8), (9) and their variations are also given in Koshy’s book [19], we refer to the new identities that we prove in this article as “Quaternionic Generalizations of Everman and Koshy”. Using the Binet formulas given in (6), Cassini’s and Catalan’s identities can also be generalized. Theorem 1. (Cassini’s Identity) For all n ∈ Z, the following identity holds Un−1Un+1 − U2 n = −(−q)n−1. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 7 of 22 Proof. Using the Binet formula and the relation αβ = −q, we have Un−1Un+1 − U2 n = (αn−1 − βn−1)(αn+1 − βn+1) (α − β)2 − (αn − βn)2 (α − β)2 = α2n + β2n − αn−1βn+1 − αn+1βn−1 (α − β)2 − α2n + β2n − 2αnβn (α − β)2 = −αn+1βn−1 − αn−1βn+1 + 2αnβn (α − β)2 = −(αβ)n−1 α2 + β2 − 2αβ (α − β)2 = −(αβ)n−1 = −(−q)n−1. Theorem 2. For all n ∈ Z, the following identity holds Vn−1Vn+1 − V 2 n = (−q)n−1∆. Proof. Using the Binet formula and identities αβ = −q, α − β = √ ∆, we obtain Vn−1Vn+1 − V 2 n = (αn−1 + βn−1)(αn+1 + βn+1) − (αn + βn)2 = (α2n + β2n + αn−1βn+1 + αn+1βn−1) − (α2n + β2n + 2αnβn) = αn+1βn−1 + αn−1βn+1 − 2αnβn = (αβ)n−1(α2 + β2 − 2αβ) = (αβ)n−1(α − β)2 = (−q)n−1∆. Theorem 3. (Catalan’s Identity) For all n, r ∈ Z, it follows that Un−rUn+r − U2 n = −(−q)n−rU2 r . Proof. Using the Binet formula and αβ = −q, we have Un−rUn+r − U2 n = (αn−r − βn−r)(αn+r − βn+r) (α − β)2 − (αn − βn)2 (α − β)2 = α2n + β2n − αn−rβn+r − αn+rβn−r (α − β)2 − α2n + β2n − 2αnβn (α − β)2 = −αn+rβn−r − αn−rβn+r + 2αnβn (α − β)2 = −(αβ)n−r α2r + β2r − 2αrβr (α − β)2 = −(αβ)n−r ( αr − βr α − β )2 = −(αβ)n−r(Ur)2 = −(−q)n−rU2 r . B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 8 of 22 Theorem 4. For all n, r ∈ Z, Vn−rVn+r − V 2 n = (−q)n−r∆U2 r . Proof. Considering the Binet formula, identities αβ = −q and α − β = √ ∆, we have Vn−rVn+r − V 2 n = (αn−r + βn−r)(αn+r + βn+r) − (αn + βn)2 = (α2n + β2n + αn−rβn+r + αn+rβn−r) − (α2n + β2n + 2αnβn) = αn+rβn−r + αn−rβn+r − 2αnβn = (αβ)n−r(α2r + β2r − 2αrβr) = (αβ)n−r(αr − βr)2 = (αβ)n−r ( αr − βr α − β )2 (α − β)2 = (αβ)n−rU2 r (α − β)2 = (−q)n−r∆U2 r . Corollary 1. For all n, r ∈ Z, we have Vn−rVn+r − V 2 n = −∆(Un−rUn+r − U2 n). Proof. Follows directly from Theorem 3 and Theorem 4. Şiar and Keskin [27] established several identities using the matrices [ p q 1 0 ] and [ Un+1 qUn Un qUn−1 ] , and proved the following results V 2 n − (p2 + 4q)U2 n = 4(−q)n, ∆UmUn = Vm+n − (−q)nVm−n, (−q)nVm−n = Um+1Vn − Vn+1Um, UrUm+n+r = Um+rUn+r − (−q)rUmUn, UrUm+n−r = UmUn − (−q)rUm−rUn−r, UrUm+n = UmUn+r − (−q)rUm−rUn, VrVm+n+r = Vm+rVn+r + (−q)r∆UmUn, VrVm+n−r = (−q)rVm−rVn−r + ∆UmUn, VrUm+n = UnVm+r + (−q)rVn−rUm, and more. Now we will prove other identities as follows. Theorem 5. For all n, r ∈ Z, we have U2 n+r − q2rU2 n−r = U2nU2r. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 9 of 22 Proof. U2 n+r − q2rU2 n−r = ( αn+r − βn+r α − β )2 − (−q)2r ( αn−r − βn−r α − β )2 = α2n+2r + β2n+2r − 2 (αβ)n+r (α − β)2 − (αβ)2r α2n−2r + β2n−2r − 2 (αβ)n−r (α − β)2 = [ α2n+2r + β2n+2r − 2 (αβ)n+r ] − (αβ)2r [ α2n−2r + β2n−2r − 2 (αβ)n−r ] (α − β)2 = α2n+2r + β2n+2r − α2nβ2r − α2rβ2n (α − β)2 = α2n+2r − α2nβ2r + β2n+2r − α2rβ2n (α − β)2 = α2n ( α2r − β2r ) + β2n ( β2r − α2r ) (α − β)2 = α2n ( α2r − β2r ) − β2n ( α2r − β2r ) (α − β)2( α2n − β2n ) ( α2r − β2r ) (α − β)2 = ( α2n − β2n ) α − β ( α2r − β2r ) α − β = U2nU2r. Theorem 6. For all n, r ∈ Z, it follows that V 2 n+r − q2rV 2 n−r = ∆U2nU2r. Proof. V 2 n+r − q2rV 2 n−r = ( αn+r + βn+r )2 − (−q)2r (αn−r + βn−r)2 = [ α2n+2r + β2n+2r + 2 (αβ)n+r ] − (αβ)2r [ α2n−2r + β2n−2r + 2 (αβ)n−r ] = α2n+2r + β2n+2r − α2nβ2r − α2rβ2n = α2n ( α2r − β2r ) + β2n ( β2r − α2r ) = α2n ( α2r − β2r ) − β2n ( α2r − β2r ) = ( α2n − β2n ) ( α2r − β2r ) = ( α2n − β2n ) α − β ( α2r − β2r ) α − β (α − β)2 = ∆U2nU2r. The following result is a direct consequence of Theorem 5 and Theorem 6. Corollary 2. For all n, r ∈ Z, it follows that V 2 n+r − q2rV 2 n−r = ∆ [ U2 n+r − q2rU2 n−r ] . B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 10 of 22 3. Generalized Fibonacci and Lucas Quaternions Quaternions are four-dimensional hypercomplex numbers introduced by William Rowan Hamilton in the 19th century and used to represent transformations in three-dimensional space [28]. Quaternions, whose coefficients are Fibonacci and Lucas numbers, have also become an interesting sequence with many generalizations over time. Let us start by giving the fundamental definition of a quaternion. Definition 5. Let a, b, c, d ∈ R and i2 = −1, j2 = −1, k2 = −1, with the multiplication rules ij = k = −ki, jk = i = −kj, and ki = j = −ik. A hypercomplex number of the form q = a + bi + cj + dk is called a quaternion [28]. The elements 1, i, j, and k are called the basis or characteristic elements of the quaternion. Definition 6. Let a1, a2, a3, a4, b1, b2, b3, b4 ∈ R. For two quaternions a = a1 + a2i + a3j + a4k, b = b1 + b2i + b3j + b4k, the addition, subtraction, and multiplication operations in the quaternion algebra are de- fined as follows: a + b = (a1 + a2i + a3j + a4k) + (b1 + b2i + b3j + b4k) = (a1 + b1) + (a2 + b2)i + (a3 + b3)j + (a4 + b4)k a − b = (a1 + a2i + a3j + a4k) − (b1 + b2i + b3j + b4k) = (a1 − b1) + (a2 − b2)i + (a3 − b3)j + (a4 − b4)k ab = (a1 + a2i + a3j + a4k)(b1 + b2i + b3j + b4k) = a1(b1 + b2i + b3j + b4k) + a2i(b1 + b2i + b3j + b4k) + a3j(b1 + b2i + b3j + b4k) + a4k(b1 + b2i + b3j + b4k) = (a1b1 + a1b2i + a1b3j + a1b4k) + (a2b1i + a2b2ii + a2b3ij + a2b4ik) + (a3b1j + a3b2ji + a3b3jj + a3b4jk) + (a4b1k + a4b2ki + a4b3kj + a4b4kk) = (a1b1 − a2b2 − a3b3 − a4b4) + i(a1b2 + a2b1 + a3b4 − a4b3) + j(a1b3 + a3b1 − a2b4 + a4b2) + k(a1b4 + a4b1 + a2b3 − a3b2). The quaternion q̄ = a − bi − cj − dk is called the conjugate of the quaternion q = a + bi + cj + dk. The norm of q is defined by N(q) = ∥q∥ = √ qq̄ = √ a2 + b2 + c2 + d2, [29–31]. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 11 of 22 Horadam, in [32], defined Fibonacci and Lucas quaternions as Qn = Fn + Fn+1i + Fn+2j + Fn+3k, Kn = Ln + Ln+1i + Ln+2j + Ln+3k, for all n ∈ Z, where Fn and Ln are Fibonacci and Lucas numbers, respectively. In this section, we define the (p, q) generalizations of the Fibonacci and Lucas quater- nions and examine their properties. Definition 7. Let Un be the n-th generalized Fibonacci number. For all n ∈ N, quaternions of the form Qn = Un + Un+1i + Un+2j + Un+3k are called generalized Fibonacci quaternions. Let Vn be the n-th generalized Lucas number. For all n ∈ N, quaternions of the form Kn = Vn + Vn+1i + Vn+2j + Vn+3k are called generalized Lucas quaternions [28, 29, 31, 33]. Binet Fomulas for generalized Fibonacci and Lucas quaternions will be given in The- orem 7. Theorem 7. ([33]) Let α̂ = 1 + αi + α2j + α3k and β̂ = 1 + βi + β2j + β3k, then the generalized Fibonacci and Lucas quaternions are given by Qn = αnα̂ − βnβ̂ α − β and Kn = αnα̂ + βnβ̂, for all n ∈ N. Proof. Using (6) in the expression Qn = Un + Un+1i + Un+2j + Un+3k, we obtain Qn = αn − βn α − β + αn+1 − βn+1 α − β i + αn+2 − βn+2 α − β j + αn+3 − βn+3 α − β k = (αn + αn+1i + αn+2j + αn+3k) − (βn + βn+1i + βn+2j + βn+3k) α − β = αn(1 + αi + α2j + α3k) − βn(1 + βi + β2j + β3k) α − β = αnα̂ − βnβ̂ α − β . Similarly, using (6) in the equation Kn = Vn + Vn+1i + Vn+2j + Vn+3k, we get Kn = (αn + βn) + (αn+1 + βn+1)i + (αn+2 + βn+2)j + (αn+3 + βn+3)k = αn(1 + αi + α2j + α3k) + βn(1 + βi + β2j + β3k) B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 12 of 22 = αnα̂ + βnβ̂. The generalized negative-indexed Fibonacci and Lucas quaternions were defined by Iakin in [31] as: Q−n = U−n +U−n+1i+U−n+2j +U−n+3k and K−n = V−n +V−n+1i+V−n+2j +V−n+3k for all n ∈ N. For example, Q−1 = U−1+U0i+U1j+U2k = 1 q +0i+j+pk, K−1 = V−1+V0i+V1j+V2k = −p q +2i+pj+(p2+2q)k. According to Iyer [34], the following properties involving the Fibonacci and Lucas quaternions hold. Here, Qn and Kn are the Fibonacci and Lucas quaternions, not their generalizations. Qn − iQn+1 − jQn+2 − kQn+3 = Ln+3, Q2 n−1 + Q2 n = 2Q2n−1 − 3L2n+2, Q2 n+1 − Q2 n−1 = QnKn = (2Q2n − 3L2n+3) + 2(−1)n+1(Q0 − 3k), Qn−2Qn−1 + QnQn+1 = 6FnQn−1 − 9F2n+2 + 2(−1)n+1(Q−1 − 3k), Qn−1Qn+3 − Q2 n+1 = (−1)n[2 + 4i + 3j + k], Qn−1Qn+1 − Qn−2Qn+2 = (−1)n[2K0 − k] + 4(−1)n+1[Q0 − 2k], Qn−3Qn−2 + QnQn+1 = 4Q2n−2 − 6L2n+1, Q2 n−1 + Q2 n+1 = 6Fn+1Qn−1 − 9F2n+3 + 2(−1)nQ−2, Qn+r + (−1)rQn−r = QnLr, Qn+1−rQn+1+r − Q2 n+1 = (−1)n−r[F 2 r K0 + F2r(Q0 − 3r)], Qn+rLn+r = Q2n+2r + (−1)n+rQ0, Qn−rLn−r = Q2n−2r + (−1)n+rQ0, Qn+rLn+r + Qn−rLn−r = Q2nL2r + 2(−1)n+rQ0, Qn+rLn+r − Qn−rLn−r = F2rK2n, Qn+rLn−r = Q2n + (−1)n−rQ2r. Before presenting the Catalan’s identity for generalized Fibonacci and Lucas quater- nions, we will first prove some preliminary lemmas. Lemma 1. Let α̂ = 1 + αi + α2j + α3k, β̂ = 1 + βi + β2j + β3k, and define A = K0 − (1 − q)(1 + q2), B = (−q)i + (−p)j + k, then the following identities hold α̂β̂ = A + qB √ ∆, (10) β̂α̂ = A − qB √ ∆. (11) B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 13 of 22 Proof. Using quaternion multiplication and the identity αβ = −q, we compute α̂β̂ = (1 + αi + α2j + α3k)(1 + βi + β2j + β3k) = [2+(α+β)i+(α2+β2)j+(α3+β3)k]−[1+αβ+α2β2+α3β3]−αβ(α−β)[αβi−(α+β)j+k] = K0 − (1 − q)(1 + q2) + q √ ∆[(−q)i + (−p)j + k] = A + qB √ ∆, which completes the proof. Similarly, it can be shown that β̂α̂ = A − qB √ ∆. Lemma 2. Let α̂ = 1 + αi + α2j + α3k, β̂ = 1 + βi + β2j + β3k, and r ∈ N. Then the following identities hold: α̂β̂βr − β̂α̂αr α − β = qBVr − AUr, (12) α̂β̂βr + β̂α̂αr = AVr − q∆BUr, (13) where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. Using equations (10) and (11), we get α̂β̂βr − β̂α̂αr α − β = (A + qB √ ∆)βr − (A − qB √ ∆)αr α − β = qB √ ∆(αr + βr) − A(αr − βr) α − β = qBVr − AUr, which proves the equation (12). Now considering the equations (10) and (11) we have α̂β̂βr + β̂α̂αr = (A + qB √ ∆)βr + (A − qB √ ∆)αr = A(αr + βr) − qB √ ∆(αr − βr) = AVr − q∆BUr, which proves the equation (13). Lemma 3. Let α̂ = 1 + αi + α2j + α3k, β̂ = 1 + βi + β2j + β3k. Then it follows that α̂β − β̂α = (−q) √ ∆Q−r, (14) α̂βr − β̂αr = √ ∆(−q)rQ−r, (15) for all r ∈ Z. Proof. Using the Binet formula, we achive that α̂β − β̂α = αβ ( α̂ α − β̂ β ) = (−q) √ ∆Q−r, α̂βr − β̂αr = (αβ)r ( α̂ αr − β̂ βr ) = √ ∆(−q)rQ−r. Now we can prove the generalized product differences of Fibonacci and Lucas quater- nions in Theorem 8 and Theorem 9. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 14 of 22 Theorem 8. For all n, r ∈ Z, it follows that Qn−rQn+r − Q2 n = (−q)n−rUr[qBVr − AUr], where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. Using the Binet formula and (12), we obtain Qn−rQn+r − Q2 n = ( α̂αn−r − β̂βn−r α − β )( α̂αn+r − β̂βn+r α − β ) − ( α̂αn − β̂βn α − β )2 = α̂2α2n + β̂2β2n − α̂β̂αn−rβn+r − β̂α̂αn+rβn−r (α − β)2 − α̂2α2n + β̂2β2n − 2α̂β̂αnβn (α − β)2 = −α̂β̂αn−rβn+r − β̂α̂αn+rβn−r + 2α̂β̂αnβn (α − β)2 = (αβ)n (α − β)2 [ α̂β̂ ( αr − βr αr ) + β̂α̂ ( βr − αr βr )] = (αβ)n(αr − βr) (α − β)2 ( α̂β̂ αr − β̂α̂ βr ) = (αβ)n(αr − βr) (α − β)2 ( α̂β̂βr − β̂α̂βr (αβ)r ) = (αβ)n−r · (αr − βr) (α − β) · α̂β̂βr − β̂α̂βr (α − β) = (αβ)n−r · αr − βr α − β · ( α̂β̂βr − β̂α̂βr α − β ) = (αβ)n−rUr · (qBVr − AUr) = (−q)n−rUr[qBVr − AUr], as desired. Theorem 9. For all n, r ∈ Z, it follows that Kn−rKn+r − K2 n = −(−q)n−rUr∆[qBVr − AUr], where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. Using the Binet formula and equation (14), we have: Kn−rKn+r − K2 n = (α̂αn−r + β̂βn−r)(α̂αn+r + β̂βn+r) − (α̂αn + β̂βn)2 = α̂2α2n + β̂2β2n + α̂β̂αn−rβn+r + β̂α̂αn+rβn−r − ( α̂2α2n + β̂2β2n + 2α̂β̂αnβn ) = −α̂β̂αn−rβn+r − β̂α̂αn+rβn−r + 2α̂β̂αnβn = −(−q)n−rUr∆[qBVr − AUr]. As a consequence of Theorem 8 and Theorem 9, we can give Corollary 3. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 15 of 22 Corollary 3. For all n, r ∈ Z, it follows that Kn−rKn+r − K2 n = −∆[Qn−rQn+r − Q2 n]. Theorem 10. For all n, r ∈ Z, it follows that Q2 n+r − q2rQ2 n−r = U2r [2Q2n − (U2n + U2n+2 + U2n+4 + U2n+6)] . Proof. Q2 n+r − q2rQ2 n−r = ( α̂αn+r − β̂βn+r α − β )2 − (−q)2r ( α̂αn−r − β̂βn−r α − β )2 = α̂α̂α2n+2r + β̂β̂β2n+2r − (αβ)n+r ( α̂β̂ + β̂α̂ ) (α − β)2 −(αβ)2r α̂α̂α2n−2r + β̂β̂β2n−2r − (αβ)n−r ( α̂β̂ + β̂α̂ ) (α − β)2 = [ α̂α̂α2n+2r + β̂β̂β2n+2r − (αβ)n+r ( α̂β̂ + β̂α̂ )] − (αβ)2r [ α̂α̂α2n−2r + β̂β̂β2n−2r − (αβ)n−r ( α̂β̂ + β̂α̂ )] (α − β)2 = α̂α̂α2n+2r + β̂β̂β2n+2r − α̂α̂α2nβ2r − β̂β̂α2rβ2n (α − β)2 = α̂α̂ ( α2n+2r − α2nβ2r ) + β̂β̂ ( β2n+2r − α2rβ2n ) (α − β)2 = α̂α̂α2n ( α2r − β2r ) + β̂β̂β2n ( β2r − α2r ) (α − β)2 = α̂α̂α2n ( α2r − β2r ) − β̂β̂β2n ( α2r − β2r ) (α − β)2 = α2r − β2r α − β α̂α̂α2n − β̂β̂β2n α − β = U2r α̂α̂α2n − β̂β̂β2n α − β = U2r α2n ( 2α̂ − 1 − α2 − α4 − α6)− β2n ( 2β̂ − 1 − β2 − β4 − β6 ) α − β = U2r 2 ( α̂α2n − β̂β2n ) − ( α2n − β2n ) − ( α2n+2 − β2n+2)− ( α2n+4 − β2n+4)− ( α2n+6 − β2n+6) α − β = U2r [ 2 α̂α2n − β̂β2n α − β − α2n − β2n α − β − α2n+2 − β2n+2 α − β − α2n+4 − β2n+4 α − β − α2n+6 − β2n+6 α − β ] = U2r [2Q2n − (U2n + U2n+2 + U2n+4 + U2n+6)] . B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 16 of 22 Theorem 11. For all n, r ∈ Z, it follows that K2 n+r − q2rK2 n−r = ∆U2r [2Q2n − (U2n + U2n+2 + U2n+4 + U2n+6)] . Proof. K2 n+r − q2rK2 n−r = ( α̂αn+r + β̂βn+r )2 − (−q)2r ( α̂αn−r + β̂βn−r )2 = [ α̂α̂α2n+2r + β̂β̂β2n+2r + (αβ)n+r ( α̂β̂ + β̂α̂ )] −(αβ)2r [ α̂α̂α2n−2r + β̂β̂β2n−2r + (αβ)n−r ( α̂β̂ + β̂α̂ )] = α̂α̂α2n+2r+β̂β̂β2n+2r−α̂α̂α2nβ2r−β̂β̂α2rβ2n = α̂α̂ ( α2n+2r − α2nβ2r ) +β̂β̂ ( β2n+2r − α2rβ2n ) = α̂α̂α2n ( α2r − β2r ) + β̂β̂β2n ( β2r − α2r ) = α̂α̂α2n ( α2r − β2r ) − β̂β̂β2n ( α2r − β2r ) = [ α2r − β2r ] [ α̂α̂α2n − β̂β̂β2n ] = [ α2r − β2r ] [ α2n ( 2α̂ − 1 − α2 − α4 − α6 ) − β2n ( 2β̂ − 1 − β2 − β4 − β6 )] = [ α2r − β2r ] [ 2 ( α̂α2n − β̂β2n ) − ( α2n − β2n ) − ( α2n+2 − β2n+2 ) − ( α2n+4 − β2n+4 ) − ( α2n+6 − β2n+6 )] = (α − β)2 [ α2r − β2r α − β ] [ 2 α̂α2n − β̂β2n α − β − α2n − β2n α − β − α2n+2 − β2n+2 α − β − α2n+4 − β2n+4 α − β − α2n+6 − β2n+6 α − β ] = ∆U2r [2Q2n − (U2n + U2n+2 + U2n+4 + U2n+6)] As a consequence of Theorem 10 and Theorem 11, we can give Corollary 4. Corollary 4. For all n, r ∈ Z, it follows that K2 n+r − q2rK2 n−r = ∆ [ Q2 n+r − q2rQ2 n−r ] . Theorem 12. For all n, r ∈ Z, it follows that K2 n − ∆Q2 n = 4 (−q)n A, where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 17 of 22 Proof. K2 n − ∆Q2 n = ( α̂αn + β̂βn )2 − ∆ ( α̂αn − β̂βn α − β )2 = [ α̂α̂α2n + β̂β̂β2n + (αβ)n ( α̂β̂ + β̂α̂ )] − ∆  α̂α̂α2n + β̂β̂β2n − (αβ)n ( α̂β̂ + β̂α̂ ) (α − β)2  = [ α̂α̂α2n + β̂β̂β2n + (αβ)n ( α̂β̂ + β̂α̂ )] − [ α̂α̂α2n + β̂β̂β2n − (αβ)n ( α̂β̂ + β̂α̂ )] = 2 (αβ)n (2A) = 4 (−q)n A. Theorem 13. For all m, n ∈ Z, it follows that K2mK2n − ∆Q2 m+n = (−q)2n Vm−n [AVm−n + Bq∆Um−n] , where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. K2mK2n−∆Q2 m+n = ( α̂α2m + β̂β2m ) ( α̂α2n + β̂β2n ) −∆ ( α̂αm+n − β̂βm+n α − β )2 = [ α̂α̂α2m+2n + β̂β̂β2m+2n + α̂β̂α2mβ2n + β̂α̂α2nβ2m ] − [ α̂α̂α2m+2n + β̂β̂β2m+2n − (αβ)m+n ( α̂β̂ + β̂α̂ )] = α̂β̂ ( α2mβ2n + αm+nβm+n ) + β̂α̂ ( α2nβ2m + αm+nβm+n ) = αm+nβ2nα̂β̂ (αm−n + βm−n) + α2nβm+nβ̂α̂ (αm−n + βm−n) = (αm−n + βm−n) [ αm+nβ2nα̂β̂ + α2nβm+nβ̂α̂ ] = (αβ)2n (αm−n + βm−n) [ αm−nα̂β̂ + βm−nβ̂α̂ ] = (αβ)2n (αm−n + βm−n) [ αm−n ( A + Bq √ ∆ ) + βm−n ( A − Bq √ ∆ )] = (αβ)2n (αm−n + βm−n) [ A (αm−n + βm−n) + Bq √ ∆ (αm−n − βm−n) ] = (−q)2n Vm−n [AVm−n + Bq∆Um−n] . Theorem 14. For all m, n ∈ Z, it follows that K2mK2n − K2 m+n = (−q)2n ∆Um−n [AUm−n + BqVm−n] , where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. K2mK2n − K2 m+n = ( α̂α2m + β̂β2m ) ( α̂α2n + β̂β2n ) − ( α̂αm+n + β̂βm+n )2 = [ α̂α̂α2m+2n + β̂β̂β2m+2n + α̂β̂α2mβ2n + β̂α̂α2nβ2m ] − [ α̂α̂α2m+2n + β̂β̂β2m+2n + (αβ)m+n ( α̂β̂ + β̂α̂ )] = α̂β̂ ( α2mβ2n − αm+nβm+n ) + β̂α̂ ( α2nβ2m − αm+nβm+n ) B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 18 of 22 = αm+nβ2nα̂β̂ (αm−n − βm−n) + α2nβm+nβ̂α̂ (βm−n − αm−n) = (αm−n − βm−n) [ αm+nβ2nα̂β̂ − α2nβm+nβ̂α̂ ] = (αβ)2n (αm−n − βm−n) [ αm−nα̂β̂ − βm−nβ̂α̂ ] = (αβ)2n (αm−n − βm−n) [ αm−n ( A + Bq √ ∆ ) − βm−n ( A − Bq √ ∆ )] = (αβ)2n (αm−n − βm−n) [ A (αm−n − βm−n) + Bq √ ∆ (αm−n + βm−n) ] = (−q)2n √ ∆Um−n [ A √ ∆Um−n + Bq √ ∆Vm−n ] = (−q)2n ∆Um−n [AUm−n + BqVm−n] . Let us generalize Everman’s product difference of Fibonacci identities to generalized Fibonacci quaternions in Theorem 15. Theorem 15. For all n, k and h ∈ Z, it follows that Qn+hQn+k − QnQn+k+h = (−q)n Uh [AUk − qBVk] , where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. Qn+hQn+k−QnQn+k+h = αn+hα̂ − βn+hβ̂ α − β αn+kα̂ − βn+kβ̂ α − β −αnα̂ − βnβ̂ α − β αn+h+kα̂ − βn+h+kβ̂ α − β = [ αn+hα̂ − βn+hβ̂ ] [ αn+kα̂ − βn+kβ̂ ] − [ αnα̂ − βnβ̂ ] [ αn+h+kα̂ − βn+h+kβ̂ ] (α − β)2 = [ α2n+h+k α̂α̂ − αn+hβn+k α̂β̂ − αn+kβn+h β̂α̂ + β2n+h+k β̂β̂ ] − [ α2n+h+k α̂α̂ − αnβn+h+k α̂β̂ − αn+h+kβn β̂α̂ + β2n+h+k β̂β̂ ] (α − β)2 = [ −αn+hβn+kα̂β̂ − αn+kβn+hβ̂α̂ ] − [ −αnβn+h+kα̂β̂ − αn+h+kβnβ̂α̂ ] (α − β)2 = −αn+hβn+kα̂β̂ − αn+kβn+hβ̂α̂ + αnβn+h+kα̂β̂ + αn+h+kβnβ̂α̂ (α − β)2 = αnβn [ α̂β̂ ( βh+k − αhβk ) + β̂α̂ ( αh+k − αkβh )] (α − β)2 = αnβn [ −βkα̂β̂ ( αh − βh ) + αkβ̂α̂ ( αh − βh )] (α − β)2 = αnβn ( αh − βh ) [ αkβ̂α̂ − βkα̂β̂ ] (α − β)2 = αnβn ( αh − βh ) [ αk ( A − qB √ ∆ ) − βk ( A + qB √ ∆ )] (α − β)2 B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 19 of 22 = (αβ)n αh − βh α − β A ( αk − βk ) α − β − q √ ∆B αk + βk α − β  = (−q)n Uh [AUk − qBVk] . Now we will generalize Everman’s product difference of Fibonacci identity to general- ized Lucas quaternions in Theorem 16. Theorem 16. For all n, k and h ∈ Z, it follows that Kn+hKn+k − KnKn+k+h = ∆ (−q)n Uh [qBVk − AUk] , where A = K0 − (1 − q)(1 + q2) and B = (−q)i + (−p)j + k. Proof. Kn+hKn+k−KnKn+k+h = [ αn+hα̂ + βn+hβ̂ ] [ αn+kα̂ + βn+kβ̂ ] − [ αnα̂ + βnβ̂ ] [ αn+h+kα̂ + βn+h+kβ̂ ] = [ α2n+h+kα̂α̂ + αn+hβn+kα̂β̂ + αn+kβn+hβ̂α̂ + β2n+h+kβ̂β̂ ] − [ α2n+h+kα̂α̂ + αnβn+h+kα̂β̂ + αn+h+kβnβ̂α̂ + β2n+h+kβ̂β̂ ] = [ αn+hβn+kα̂β̂ + αn+kβn+hβ̂α̂ ] − [ αnβn+h+kα̂β̂ + αn+h+kβnβ̂α̂ ] = αn+hβn+kα̂β̂ + αn+kβn+hβ̂α̂ − αnβn+h+kα̂β̂ − αn+h+kβnβ̂α̂ = αnβn [ α̂β̂ ( αhβk − βh+k ) + β̂α̂ ( αkβh − αh+k )] = αnβn [ α̂β̂βk ( αh − βh ) + β̂α̂αk ( βh − αh )] = (αβ)n ( αh − βh ) [ α̂β̂βk − β̂α̂αk ] = (αβ)n ( αh − βh ) [ βk ( A + qB √ ∆ ) − αk ( A − qB √ ∆ )] = (αβ)n ( αh − βh ) [ qB √ ∆ ( αk + βk ) − A ( αk − βk )] = ∆ (αβ)n ( αh − βh α − β )qB √ ∆ ( αk + βk ) − A ( αk − βk ) α − β  = ∆ (−q)n Uh [qBVk − AUk]. As a consequence of Theorem 15 and Theorem 16, we can give Corollary 5. Corollary 5. For all n, k and h ∈ Z, it follows that Kn+hKn+k − KnKn+k+h = −∆ [Qn+hQn+k − QnQn+k+h] . B. Demirtürk, N. Topal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6492 20 of 22 4. Conclusion This paper presents new identities for both theoretical mathematical theories and var- ious applied fields, by integrating generalized Fibonacci and Lucas sequences with quater- nion structures. Known identities of Fibonacci and Lucas sequences, such as the classical Binet formulas, Cassini’s, Catalan’s and d’Ocagne’s identities are generalized, as well as new identities, are rederived through quaternion extensions. Therefore, this paper demon- strates the interaction between number sequences and quaternions by extending some fundamental equations in the existing literature to high-dimensional hypercomplex struc- tures. In particular, the original part of the paper is the reconstruction of the classical results of researchers such as Koshy and Everman by means of generalized Fibonacci and Lucas sequences and the systematic presentation of the quaternion forms of these equa- tions and product differences generalizations. 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