fundamental journal of mathematics and applications VOLUME IV ISSUE I www.dergipark.org.tr/en/pub/fujma ISSN 2645-8845 VOLUME 4 ISSUE 1 March 2021 ISSN 2645-8845 www.dergipark.org.tr/en/pub/fujma FUNDAMENTAL JOURNAL OF MATHEMATICS AND APPLICATIONS Editors in Chief Mahmut Akyi˘git Merve Ilkhan˙ Kara Department of Mathematics, Department of Mathematics, Faculty of Science and Arts, Sakarya University, Faculty of Science and Arts, D¨uzce University, Sakarya-TURK¨ IYE˙ D¨uzce-TURK¨ IYE˙ [email protected] [email protected] Soley Ersoy Department of Mathematics, Faculty of Science and Arts, Sakarya University, Sakarya-TURK¨ IYE˙ [email protected] Managing Editor Fuat Usta Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-TURK¨ IYE˙ [email protected] Editorial Board Stoil I. Ivanov Slavica Iveli´cBradanovi´c University of Plovdiv Paisii Hilendarski University of Split BULGARIA CROATIA Murat Tosun Mohammad Saeed KHAN Sakarya University Sultan Qaboos University TURK¨ IYE˙ OMAN Vladimir Vladicic Emrah Evren Kara University East Sarajevo D¨uzce University BOSNIA and HERZEGOVINA TURK¨ IYE˙ T¨ulay Kesemen Mohammad Mursaleen Karadeniz Technical University Aligarh Muslim University TURK¨ IYE˙ INDIA i Zlatko Pavi´c Sidney Allen Morris University of Osijek Federation University CROATIA AUSTRALIA Ramazan Kama Naoyuki Ishimura Siirt University Chuo University TURK¨ IYE˙ JAPAN Syed Abdul Mohiuddine Figen Oke¨ King Abdulaziz University Trakya University SAUDI ARABIA TURK¨ IYE˙ Ismet˙ Altınta¸s Ay¸se Yılmaz Ceylan Kyrgyz-Turkish Manas University Akdeniz University KYRGYZSTAN TURK¨ IYE˙ Editorial Secretariat Editorial Secretariat Bahar Do˘ganYazıcı Department of Mathematics Bilecik S¸eyh Edebali University Bilecik-TURK¨ IYE˙ ii Contents 1 Unrestricted Fibonacci and Lucas Quaternions Ahmet Da¸sdemir, G¨oksal Bilgici 1-9 2 Fibonacci Elliptic Biquaternions Kahraman Esen Ozen,¨ Murat Tosun 10-16 3 Faber Polynomial Expansion for a New Subclass of Bi-univalent Functions Endowed with (p,q) Calculus Operators Om P. Ahuja, Asena C¸etinkaya 17-24 4 Construction of Degenerate q-Daehee Polynomials with Weight α and its Applications Serkan Aracı 25-32 5 Discrete Networked Dynamic Systems with Eigen-Spectrum Gap: Analysis and Performance Magdi S. Mahmoud, Bilal J. Karaki 33-44 6 I-Limit and I-Cluster Points for Functions Defined on Amenable Semigroups U˘gur Ulusu, Fatih Nuray, Erdin¸cD¨undar 45-48 7 Circulant m-Diagonal Matrices Associated With Chebyshev Polynomials Ahmet Otele¸s¨ 49-58 8 Recurrence Relations for Knot Polynomials of Twist Knots Kemal Ta¸sk¨opr¨u, Zekiye S¸evval Sinan 59-66 iii Fundamental Journal of Mathematics and Applications, 4 (1) (2021) 1-9 Research Article Fundamental Journal of Mathematics and Applications Journal Homepage: www.dergipark.gov.tr/fujma ISSN: 2645-8845 doi: 10.33401/fujma.752758 Unrestricted Fibonacci and Lucas Quaternions Ahmet Das¸demir1* and Goksal¨ Bilgici2 1Department of Mathematics, Faculty of Arts and Sciences, Kastamonu University, Kastamonu, Turkey 2Department of Elementary Mathematics Education, Faculty of Education, Kastamonu University, Kastamonu, Turkey *Corresponding author Article Info Abstract Keywords: Binet’s formula, Catalan’s Many quaternion numbers associated with Fibonacci and Lucas numbers or even their identity, Fibonacci quaternion, Gener- generalizations have been defined and widely discussed so far. In all the studies, the ating function, Lucas quaternion coefficients of these quaternions have been selected from consecutive terms of these numbers. 2010 AMS: 11B39, 15B36, 11R52 In this study, we define other generalizations for the usual Fibonacci and Lucas quaternions. Received: 14 June 2020 We also present some properties, including the Binet’s formulas and d’Ocagne’s identities, Accepted: 20 January 2021 for these types of quaternions. Available online: 29 January 2021 1. Introduction Since Sir William Rowan Hamilton introduced the quaternion algebra in 1843, their usage areas have developed rapidly. Due to the fact that the quaternions are encountered in many problems from elastodynamics, quantum mechanics, elasticity theory and many other fields of modern sciences, they have been studied widely. A quaternion q can be regarded as a quadruple of real numbers and is formally defined by q = q0 + iq1 + jq2 + kq3, where q0,q1,q2, and q3 are any real numbers and the standard basis 1,i, j,k satisfies { } i2 = j2 = k2 = 1,i j = ji = k, jk = k j = i,ki = ik = j. − − − − The conjugate of q is q∗ = q0 iq1 jq2 kq3 − − − and, the norm of q is Email addresses and ORCID numbers: [email protected], https://orcid.org/0000-0001-8352-2020 (A. Das¸demir), [email protected], https://orcid.org/0000-0001-9964-5578 (G. Bilgici) 2 Fundamental Journal of Mathematics and Applications 2 2 2 2 N(q)= qq∗ = q0 + q1 + q2 + q3. Fibonacci numbers are recursively defined as Fn = Fn 1 + Fn 2, − − with the initial terms F0 = 0 and F1 = 1. The Lucas numbers satisfy the same recurrence relation but with the initial terms L0 = 2 and L1 = 1. Binet’s formulas for the Fibonacci and Lucas numbers are αn β n F = − and L = αn + β n, n α β n − 1+√5 1 √5 2 respectively. Here, α = and β = − are the positive and negative roots of x x 1 = 0, respectively. 2 2 − − Horadam [1] defined Fibonacci quaternions as Qn := Fn + iFn+1 + jFn+2 + kFn+3, (1.1) where Fn is the nth term of the Fibonacci sequence. Iyer [2] gave a similar definition for Lucas quaternions by the relation Tn := Ln + iLn+1 + jLn+2 + kLn+3, (1.2) and gave their many properties, where Ln is the nth Lucas number. Halici [3] gave Binet’s formulas for the Fibonacci and Lucas quaternions as follows: αα ββ Q = − and T = αα + ββ, n α β n − where α = 1 + iα + jα2 + kα3 and β = 1 + iβ + jβ 2 + kβ 3. There exist many papers devoted to generalizations of the quaternion sequences in (1.1) and (1.2) today. For example, the references in [4]–[11] can be investigated. Note that all authors have used a known generalization of Fibonacci and Lucas numbers and have placed these numbers in coefficients of the basis vectors similar to the format given by Horadam and Iyer. In this study, we present an another perspective to the Fibonacci and Lucas quaternions. According to our approach, we define a new classes of quaternions whose the coefficients are arbitrarily selected from these splendid integers. The outline of this paper is as follows: In Section 2, we introduce the unrestricted Fibonacci and Lucas quaternions and give Binet’s formulas and the generating functions for these quaternions. Furthermore, we present certain special identities such as d’Ocagne’s identity and Catalan’s identity; in Section 3, we display many fundamental properties and some sum formulas for these quaternion families. 2. Main results Here, we present our definitions, some concepts and results. First of all, we give a definition in the following. Definition 2.1. Let p, r and s be arbitrary integers. Hence, nth unrestricted Fibonacci and Lucas quaternions are given by the relations (p,r,s) Fn := Fn + iFn+p + jFn+r + kFn+s (2.1) and (p,r,s) Ln := Ln + iLn+p + jLn+r + kLn+s, (2.2) respectively. Fundamental Journal of Mathematics and Applications 3 According to our definitions, we have the following special cases: • For p = r = s = n, the usual Fibonacci numbers are obtained: − ( n, n, n) Fn− − − = Fn • For p = 1 and r = s = n, the Gaussian Fibonacci numbers are obtained: − (1, n, n) Fn − − = Fn + iFn+1 • For p = 1, r = 2 and s = 3, the well-known Fibonacci and Lucas quaternions are obtained: (1,2,3) Fn = Qn = Fn + iFn+1 + jFn+2 + kFn+3 (1,2,3) Ln = Tn = Ln + iLn+1 + jLn+2 + kLn+3 Taking Eqs. (2.1) and (2.2) into account, we directly obtain F (p,r,s) F (p,r,s) F (p,r,s) n = n 1 + n 2 (2.3) − − and L (p,r,s) L (p,r,s) L (p,r,s) n = n 1 + n 2 , (2.4) − − by the recurrence relations of Fibonacci and Lucas numbers respectively. To make it easier to present and prove the results of the rest of paper, we now present next theorem that states Binet’s formulas for the unrestricted Fibonacci and Lucas quaternions. Theorem 2.2. (Binet’s Formula) Let n be an integer. Then the Binet’s formulas of the unrestricted Fibonacci and Lucas quaternions are n ˘ n p,r,s αα˘ ββ F ( ) = − (2.5) n α β − and (p,r,s) n n Ln = αα˘ + ββ˘ , (2.6) where α˘ = 1 + iα p + jαr + kαs and β˘ = 1 + iβ p + jβ r + kβ s. Proof. Applying Binet’s formulas of the classic Fibonacci numbers to the definition of the unrestricted Fibonacci quaternions, we obtain (p,r,s) Fn = Fn + iFn+p + jFn+r + kFn+s 1 = αn β n + i(αn+p β n+p)+ j(αn+r β n+r)+ k(αn+s β n+s) α β − − − − − 1 = αn(1 + iα p + jαr + kαs) β n(1 + iβ p + jβ r + kβ s) , α β − − and the last equation gives Eq. (2.5). The other can be proved similarly. 4 Fundamental Journal of Mathematics and Applications α˘ β˘ α˘ β˘ · ∗ ∗ α˘ 2α˘ N(α˘ ) B + √5C N(α˘ ) A √5C + 2γ − − β˘ B √5C 2β˘ N β˘ A + √5C N β˘ − − − α˘ N(α˘ ) A √5C 2α˘ N(α˘ ) B + √5C 2γ + 4 ∗ − − ∗ − − − β˘ A + √5C + 2γ N β˘ B √5C 2γ + 4 2β˘ N β˘ ∗ − − − ∗ − Table 1: The multiplicative properties of α˘ and β˘ Table 1 displays important features corresponding to the multiplication of α˘ and β˘. They will undertake important tasks in the process of proofing the next theorems.