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The Generalization of Gaussians and Leonardo’s Octonions

The Generalization of Gaussians and Leonardo’s Octonions Annales Mathematicae Silesianae 37 (2023), no. 1, 117–137 DOI: 10.2478/amsil-2023-0004 THE GENERALIZATION OF GAUSSIANS AND LEONARDO’S OCTONIONS Renata Passos Machado Vieira , Milena Carolina dos Santos Mangueira, Francisco Régis Vieira Alves, Paula Maria Machado Cruz Catarino Abstract. In order to explore the Leonardo sequence, the process of complex- ification of this sequence is carried out in this work. With this, the Gaussian and octonion numbers of the Leonardo sequence are presented. Also, the re- currence, generating function, Binet’s formula, and matrix form of Leonardo’s Gaussian and octonion numbers are defined. The development of the Gauss- ian numbers is performed from the insertion of the imaginary component i in the one-dimensional recurrence of the sequence. Regarding the octonions, the terms of the Leonardo sequence are presented in eight dimensions. Further- more, the generalizations and inherent properties of Leonardo’s Gaussians and octonions are presented. 1. Introduction Historically, the origin of the Leonardo sequence reports that this sequence was possibly created by Leonardo of Pisa (1180-1250) [2]. This fact is due to its great similarity with the Fibonacci sequence and also because the sequence Received: 11.01.2022. Accepted: 06.02.2023. Published online: 27.02.2023. (2020) Mathematics Subject Classification: 11B37, 11B39. Key words and phrases: matrix form, Binet’s formula, Gaussian, octonions, Leonardo sequence. The part of research development in Brazil had the financial support of the National Council for Scientific and Technological Development – CNPq and the Ceará Foundation for Support to Scientific and Technological Development (Funcap). The research development aspect in Portugal is financed by National Funds through FCT – Fundação para a Ciência e Tecnologia. I.P, within the scope of the UID/CED/00194/2020 project. 2023 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). 118 Renata Passos Machado Vieira et al. has the same name as the Italian mathematician. It is possible to find the mathematical evolution of this sequence in the works [3, 1, 7, 10, 8]. The Leonardo sequence satisfies the following recurrence relation: L = L + L + 1; n  2; L = L = 1: n n1 n2 0 1 On the other hand, for n + 1 we can rewrite this recurrence relation as L = L + L + 1. With this, we can add the term L , resulting in n+1 n n1 n+1 another recurrence relation. Hence, we have: L L = L + L + 1 L L 1; n n+1 n1 n2 n n1 i.e., L = 2L L ; n+1 n n2 where L = L = 1 are initial conditions. 0 1 In 1981 Harman ([4]) introduced the Gaussian numbers denoted by z = a + bi with a, b 2 Z and i = 1, which will be associated with the Leonardo sequence, improving the process of mathematical complexification of this se- quence. Regarding octonion numbers, according to Vieira, Alves, and Catarino (2020) [9], it can be said that in the work of Karatas and Halici (2017) [5] the algebra in sixteen dimensions was studied and Horadam octonions by Ho- radam sequence which is a generalization of second order recurrence relations were defined. Octonions were defined as the R numerical field, writing as ([6, 5]): 0 00 p = p + p e; 0 00 2 2 2 where p ; p 2 H = fa + a iii + a jjj + a kkk j iii = jjj = kkk = 1;ij ij ijkkk = 1; 0 1 2 3 a ; a ; a ; a 2 Rg. 0 1 2 3 For the operation of adding and multiplying between two octonions, 0 00 0 00 p = p + p e and, q = q + q e: 0 0 00 00 p + q = (p + q ) + (p + q )e; 0 0 00 00 0 00 00 0 pq = (p q q p ) + (q p + p q )e; 0 00 0 00 where q and q are the conjugates of the quaternions q and q , respectively. Therefore, O is the algebra of the octonions, on a natural basis in the space over R formed by the elements: e = 1, e = iii, e = jjj, e = kkk, e = e, e = iiie, 0 1 2 3 4 5 e = jjje, e = kkke and the multiplication of these numbers is shown in the 6 7 Table 1. The generalization of Gaussians and Leonardo’s octonions 119 Table 1. Multiplication of the octonions of O. Source: [6] 1 e e e e e e e 1 2 3 4 5 6 7 1 1 e e e e e e e 1 2 3 4 5 6 7 e e -1 e -e e -e -e e 1 1 3 2 5 4 7 6 e e -e -1 e e e -e -e 2 2 3 1 6 7 4 5 e e e -e -1 e -e e -e 3 3 2 1 7 6 5 4 e e -e -e -e -1 e e e 4 4 5 6 7 1 2 3 e e e -e e -e -1 -e e 5 5 4 7 6 1 3 2 e e e e -e -e e -1 -e 6 6 7 4 5 2 3 1 e e -e e e -e -e e -1 7 7 6 5 4 3 2 1 Thus, the following notation is used for octonions: p = p e ; s s p=0 where p is the real coefficient, with p 2 O, in format p = Re(p) + Im(p), where Re(p) = p represents the real part and, Im(p) = p e represents 0 s s s=1 the imaginary part. Therefore, in this work, it is intended to continue the mathematical evo- lution of this sequence, presenting the Gaussian and octonion numbers of Leonardo’s sequence. 2. Leonardo’s Gaussians Henceforward, Leonardo’s Gaussian numbers will be introduced, beginning complex studies around this sequence, with the insertion of an imaginary unit. Thus, their respective mathematical aspects are portrayed. Definition 2.1. For n > 0, Leonardo’s Gaussians are defined by: GL = L + iL ; n n n+1 where L = L = 1. In particular, GL = 1 + i; GL = 1 + 3i. 0 1 0 1 From the previous definition, it is easy to see that for all n > 3 and n 2 N, the recurrence formula of Leonardo’s Gaussian is given by: GL = 2GL GL ; n n1 n3 where GL = 1 + i and GL = 1 + 3i. 0 1 120 Renata Passos Machado Vieira et al. Definition 2.2. For n > 0, Fibonacci’s Gaussians are defined by: GF = F + iF ; n n n+1 where F = 0; F = 1. In particular, GF = i and GF = 1 + i. 0 1 0 1 From the previous definition, it is easy to see that for all n > 2 and n 2 N, the recurrence formula of Fibonacci’s Gaussian is given by: GF = GF + GF ; n n1 n2 where GF = i and GF = 1 + i. 0 1 Definition 2.3. Leonardo and Fibonacci’s Gaussian recurrence formula is given by: GLF = GL + GF ; n n n where n 2 Z. Theorem 2.4. The generating function of Leonardo’s Gaussians is given by: 1 + i + (1 i)(x + x ) g(GL ; x) = : (1 2x x ) Proof. Let us consider the function 2 n g(GL ; x) = GL + GL x + GL x + : : : + GL x + : : : : n 0 1 2 n Multiplying this function by 2x and x , we get 2 3 n 2xg(GL ; x) = 2GL x + 2GL x + 2GL x + : : : + 2GL x + : : : ; n 0 1 2 n1 3 3 4 5 n x g(GL ; x) = GL x + GL x + GL x + : : : + GL x + : : : : n 0 1 2 n3 Subtracting the previous equalities and after some calculations, we obtain: 3 2 (1 2x x )g(GL ; x) = GL + (GL 2GL )x + (GL 2GL )x ; n 0 1 0 2 1 3 2 (1 2x x )g(GL ; x) = 1 + i (1 i)x + (1 i)x ; 3 2 (1 2x x )g(GL ; x) = 1 + i + (1 i)(x + x ); 1 + i + (1 i)(x + x ) g(GL ; x) = : (1 2x x ) The generalization of Gaussians and Leonardo’s octonions 121 Theorem 2.5. The Binet formula of Leonardo’s Gaussians, with n 2 Z, is: n n n GL = A (1 + ir )r + B (1 + ir )r + C (1 + ir )r ; n g 1 g 2 g 3 1 2 3 3 2 where r ; r ; r are the roots of the characteristic polynomial r 2r + 1 = 0, 1 2 3 (r 1)(r 1) (r 1)(r 1) 2 3 1 3 A = ; B = ; g g (r r )(r r ) (r r )(r r ) 1 2 1 3 2 1 2 3 (r 1)(r 1) 1 2 C = : (r r )(r r ) 3 1 3 2 n n n Proof. Through the Binet’s formula GL = r + r + r and the 1 2 3 recurrence of Leonardo’s Gaussians GL = L +iL , with the initial values n n n+1 GL = 1 + i, GL = 1 + 3i and GL = 3 + 5i, it is possible to obtain the 0 1 2 following system of equations: + + = 1 + i; r + r + r = 1 + 3i; 1 2 3 2 2 2 r + r + r = 3 + 5i: 1 2 3 Solving the system, we get: (3 + 5i) + (r r )(1 + 3i) + r r (1 + i) 2 3 2 3 = ; r r r r r + r r 1 2 1 3 2 3 (3 + 5i) + (r r )(1 + 3i) + r r (1 + i) 1 3 1 3 = ; r r r r r + r r 2 3 1 2 1 3 (3 + 5i) + (r r )(1 + 3i) + r r (1 + i) 1 2 1 2 = : r + r r r r r r 1 2 1 3 2 3 Through Girard’s relations: r r r = 1; r +r +r = 2 and r r +r r + 1 2 3 1 2 3 1 2 2 3 r r = 0, it is easy to see that: 1 3 (r r r r + 1) (r 1)(r 1) 2 2 2 3 2 3 = (1 + ir )= (1 + ir ) =A (1 + ir ); 1 1 g 1 (r r )(r r ) (r r )(r r ) 1 2 1 3 1 2 1 3 (r r r r + 1) (r 1)(r 1) 1 3 1 3 1 3 = (1 + ir )= (1 + ir ) =B (1 + ir ); 2 2 g 2 (r r )(r r ) (r r )(r r ) 2 1 2 3 2 1 2 3 (r r r r + 1) (r 1)(r 1) 1 2 1 2 1 2 = (1 + ir )= (1 + ir ) =C (1 + ir ): 3 3 g 3 (r r )(r r ) (r r )(r r ) 3 1 3 2 3 1 3 2 122 Renata Passos Machado Vieira et al. Based on the work of Vieira, Mangueira, Alves and Catarino ([10]), one can establish the matrix form of Leonardo’s sequence in the complex form. Proposition 2.6. For n > 2 and n 2 N, the matrix form of Leonardo’s Gaussians is given by: 2 3 2 3 GLF GF GF 2 0 2 1 0 L n+2 6GLF 7 4 5 3 1 1 0 0 1 4 GF GF 5 1 0 n+1 GLF 1 0 0 2 GF GF 0 1 2 3 GLF GF GF 2 0 n+2 6 7 GLF = L L L GF GF n+2 n+1 n 4 5 1 0 n+1 GLF GF GF 0 1 GL GL GL = ; n+2 n+1 n where GLF = GL + GF , for n < 0. n n n Proof. By the principle of finite induction, we have for n = 2: 2 3 2 3 GLF 2 2 GF GF 2 0 2 1 0 6GLF 7 4 5 3 1 1 0 0 1 4 GF GF 5 1 0 GLF 1 0 0 GF GF 0 1 2 3 GLF GF GF 2 0 GLF 4 5 9 5 3 = GF GF 1 0 GLF GF GF 0 1 = 9GF 3GF +GLF 9GF +5GF +GLF GLF +5GF +3GF 2 0 2 0 1 2 2 0 1 GL GL GL = : 4 3 2 So, assume it is true for any n = k, with k 2 N: 2 3 2 3 GLF k 2 GF GF 2 0 2 1 0 L k+2 6 7 GLF 4 5 3 1 1 0 0 1 GL GL GL GF GF = : 4 1 0 5 k+2 k+1 k k+1 1 0 0 GLF GF GF 0 1 k The generalization of Gaussians and Leonardo’s octonions 123 Let us show that it is still valid for n = k + 1: 2 3 2 3 GLF k+1 GF GF 2 0 2 1 0 L k+3 6GLF 7 4 5 3 1 1 0 0 1 OF GF 4 1 0 5 k+2 GLF 1 0 0 2 GF GF 0 1 k+1 2 3 2 3 GLF GF GF 2 0 2 1 0 L k+3 6GLF 7 4 5 = L L L 0 0 1 GF GF 4 5 k+2 k+1 k 1 0 k+2 GLF 1 0 0 2 GF GF 0 1 k+1 2 3 GLF GF GF 2 0 k+3 6 7 GLF = L L L GF GF 4 1 0 5 k+3 k+2 k+1 k+2 GLF GF GF 0 1 k+1 = [L GF L GF + GLF L GF + L GF + GLF k+3 2 k+1 0 2 k+3 0 k+2 1 2 L GF + L GF + GLF ] k+2 0 k+1 1 2 = GL GL GL : k+3 k+2 k+1 3. The generalization of Leonardo’s Gaussians Next, the behavior of terms with non-positive integer indices of Leonardo’s Gaussians will be analyzed. Definition 3.1. For all n > 0 and n 2 N, Leonardo’s Gaussians, for non-positive integer index, are defined by the equation: GL = L e : n n+s s s=0 From the previous definition, it is easy to see that for all n > 0 and n 2 N, the recurrence formula of Leonardo’s Gaussians for non-positive integer index, is given by: GL = 2GL GL ; n n+2 n+3 where GL = 1 + i; GL = 1 + 3i and GL = 4 + 6i. 0 1 2 124 Renata Passos Machado Vieira et al. Proposition 3.2. The generating function of Leonardo’s Gaussians for non-positive integer index, is expressed by: 1 + i + (1 + i)x + (1 3i)x g(GL ; x) = : 3 2 x 2x + 1 Proof. Let us consider the function n 2 n g(GL ) = GL x = GL + GL x + GL x + : : : + GL x + : : : : n;x n 0 1 2 n n=0 2 3 Multiplying this function by 2x and x , we have: 2 2 3 4 n 2x g(GL ; x) = 2GL x + 2GL x + 2GL x + : : : + 2GL x + : : : ; n 0 1 2 n2 3 3 4 5 n x g(GL ; x) = GL x + GL x + GL x + : : : + GL x + : : : : n 0 1 2 n3 Subtracting the previous equalities and after some calculations, we obtain: 3 2 2 (x 2x + 1)g(GL ; x) = GL + GL x + (2GL + GL )x ; n 0 1 0 2 3 2 2 (x 2x + 1)g(GL ; x) = 1 + i + (1 + i)x + (1 3i)x ; 1 + i + (1 + i)x + (1 3i)x g(GL ; x) = : 3 2 x 2x + 1 Proposition 3.3. For n>0 and n2N, the generating matrix of Leonardo’s Gaussians, with a non-positive integer index, is given by: 2 3 2 3 GLF GF GF 2 0 0 0 1 n+2 6 7 GLF 4 5 3 1 1 1 0 2 GF GF 4 1 0 5 n+1 GLF 0 1 0 2 GF GF 0 1 2 3 GLF GF GF 2 0 n+2 6GLF 7 L L L GF GF n+2 n+1 n 4 1 0 5 n+1 GLF GF GF 0 1 GL GL GL = ; n+2 n+1 n where GLF = GL + GF , GF = i 1, GF = 1 and GF = i. n n n 2 1 0 Proof. Similarly to the demonstration performed in Proposition 2.6, this property can be validated.  The generalization of Gaussians and Leonardo’s octonions 125 4. Leonardo’s octonions In this section, Leonardo’s octonions will be studied, addressing their re- spective mathematical properties. Definition 4.1. For n > 0, Leonardo’s octonions are defined by: OL = L e ; n n+s s s=0 P P P 7 7 7 where OL = L e , OL = L e , OL = L e : 0 s s 1 1+s s 2 2+s s s=0 s=0 s=0 From the previous definition, it is easy to see that for all n > 3 and n 2 N, the recurrence formula of Leonardo’s octonions is given by: OL = 2OL OL ; n n1 n3 P P P 7 7 7 where OL = L e , OL = L e , OL = L e . 0 s s 1 1+s s 2 2+s s s=0 s=0 s=0 Theorem 4.2. The generating function of Leonardo’s octonions, OL , is given by: g(OL ; x) = (L + L x + L x )e : n s s2 s1 s (1 2x x ) s=0 Proof. Let us consider the function 2 n g(OL ; x) = OL + OL x + OL x + : : : + OL x + : : : : n 0 1 2 n Multiplying the function by 2x and x , we get: 2 3 n 2xg(OL ; x) = 2OL x + 2OL x + 2OL x + : : : + 2OL x + : : : ; n 0 1 2 n1 3 3 4 5 n x g(OL ; x) = OL x + OL x + OL x + : : : + OL x + : : : : n 0 1 2 n3 Subtracting the previous equalities and after some calculations, we obtain: 3 2 (1 2x x )g(OL ; x) = OL + (OL 2OL )x + (OL 2OL )x ; n 0 1 0 2 1 g(OL ; x) = (L + L x + L x )e : n s s2 s1 s (1 2x x ) s=0 126 Renata Passos Machado Vieira et al. Theorem 4.3. Binet’s formula for Leonardo’s octonions, with n 2 Z, is given by: n n n OL = r + r + r ; 1 2 3 3 2 where r ; r and r are the roots of the characteristic polynomial r 2r +1=0, 1 2 3 (r 1)(r 1) (r 1)(r 1) (r 1)(r 1) 2 3 1 3 1 2 A = ; B = ; C = ; l l l (r r )(r r ) (r x )(r r ) (r r )(r r ) 1 2 1 3 2 1 2 3 3 1 3 2 7 7 7 X X X s s s = r e ; = r e ; = r e ; ol s ol s ol s 1 2 3 s=0 s=0 s=0 = A ; = B ; = C : l ol l ol l ol n n n Proof. Through the Binet’s formula oL = r + r + r and the 1 2 3 recurrence of Leonardo’s octonions OL = L e , with the initial val- n n+s s s=0 P P P 7 7 7 ues OL = L e , OL = L e and OL = L e , it is 0 s s 1 s+1 s 2 s+2 s s=0 s=0 s=0 possible to obtain the following system of equations: + + = L e ; > s s s=0 r + r + r = L e ; 1 2 3 s+1 s s=0 > 7 > X 2 2 2 > r + r + r = L e : s+2 s : 1 2 3 s=0 Solving this system, we have: 7 7 7 X X X L e + (r r ) L e + r r L e s+2 s 2 3 s+1 s 2 3 s s s=0 s=0 s=0 = ; r r r r r + r r 1 2 1 3 2 3 7 7 7 X X X L e + (r r ) L e + r r L e s+2 s 1 3 s+1 s 1 3 s s s=0 s=0 s=0 = ; r r r r r + r r 2 3 1 2 1 3 7 7 7 X X X L e + (r r ) L e + r r L e s+2 s 1 2 s+1 s 1 2 s s s=0 s=0 s=0 = : r + r r r r r r 1 2 1 3 2 3 3 The generalization of Gaussians and Leonardo’s octonions 127 Through Girard’s relations: r x r = 1; r + r + r = 2 and r r + r r + 1 2 3 1 2 3 1 2 2 3 r r = 0, it is easy to see that: 1 3 7 7 7 X X X (r r r r + 1) (r 1)(r 1) 2 2 2 3 2 3 s s s = r e = r e = A r e ; s s l s 1 1 1 (r r )(r r ) (r r )(r r ) 1 2 1 3 1 2 1 3 s=0 s=0 s=0 7 7 7 X X X (r r r r + 1) (r 1)(r 1) 1 3 1 3 1 3 s s s = r e = r e = B r e ; s s l s 2 2 2 (r r )(r r ) (r r )(r r ) 2 1 2 3 2 1 2 3 s=0 s=0 s=0 7 7 7 X X X (r r r r + 1) (r 1)(r 1) 1 2 1 2 1 2 s s s = r e = r e = C r e : s s l s 3 3 3 (r r )(r r ) (r r )(r r ) 3 1 3 2 3 1 3 2 s=0 s=0 s=0 P P P 7 7 7 s s s Defining = r e , = r e and = r e , it is easy to ol s ol s ol s 1 2 3 s=0 s=0 s=0 see that: = A ; = B ; = C : l ol l ol l ol The matrix form of Leonardo’s octonions is based on the work of Vieira, Mangueira, Alves and Catarino ([10]), in which we found a development on the matrix form of the one-dimensional Leonardo sequence. Proposition 4.4. For n > 2 and n 2 N, the matrix form of Leonardo’s octonions is given by: 2 3 2 3 OLF OF OF 2 0 2 1 0 L n+2 6OLF 7 4 5 3 1 1 0 0 1 4 OF OF 5 1 0 n+1 OLF 1 0 0 2 OF OF 0 1 2 3 OLF OF OF 2 0 n+2 6 7 OLF = L L L OF OF n+2 n+1 n 4 5 1 0 n+1 OLF OF OF 0 1 OL OL OL = ; n+2 n+1 n where OLF = OL + OF . n n n 128 Renata Passos Machado Vieira et al. Proof. By the principle of finite induction, we have for n = 2: 2 3 2 3 OLF 2 2 OF OF 2 0 2 1 0 6 7 OLF 4 5 3 1 1 0 0 1 OF OF 4 5 1 0 OLF 1 0 0 OF OF 0 1 2 3 OLF OF OF 2 0 OLF 4 5 9 5 3 = OF OF 1 0 OLF OF OF 0 1 = 9OF 3OF +OLF 9OF +5OF +OLF OLF +5OF +3OF 2 0 2 0 1 2 2 0 1 OL OL OL = : 4 3 2 So, assume it is true for any n = k, with k 2 N: 2 3 2 3 OLF k 2 OF OF 2 0 2 1 0 L k+2 6 7 OLF 4 5 3 1 1 0 0 1 OL OL OL OF OF = : 4 1 0 5 k+2 k+1 k k+1 1 0 0 OLF OF OF 0 1 Finally, the validity for n = k + 1 is verified: 2 3 2 3 OLF k+1 OF OF 2 0 2 1 0 k+3 6OLF 7 4 5 3 1 1 0 0 1 OF OF 4 1 0 5 k+2 OLF 1 0 0 2 OF OF 0 1 k+1 2 3 2 3 OLF OF OF 2 0 2 1 0 L k+3 6OLF 7 4 5 L L L 0 0 1 OF OF k+2 k+1 k 4 1 0 5 k+2 OLF 1 0 0 2 OF OF 0 1 k+1 2 3 OLF OF OF 2 0 k+3 6OLF 7 = L L L OF OF 4 5 k+3 k+2 k+1 1 0 k+2 OLF OF OF 0 1 k+1 = [L OF L OF +OLF L OF +L OF +OLF k+3 2 k+1 0 2 k+3 0 k+2 1 2 L OF +L OF +OLF ] k+2 0 k+1 1 2 = OL OL OL : k+3 k+2 k+1 The generalization of Gaussians and Leonardo’s octonions 129 5. The generalization of Leonardo’s octonions Next, the behavior of terms with non-positive integer indices of Leonardo’s octonions will be analyzed. Definition 5.1. For all n > 0 and n 2 N, Leonardo’s octonions, for non-positive integer index, are defined by the equation: OL = L e : n n+s s s=0 From the previous definition, it is easy to see that for all n > 0 and n 2 N, the recurrence formula of Leonardo’s octonions for non-positive integer index, is given by: OL = 2OL OL ; n n+2 n+3 P P P 7 7 7 where OL = L e ; OL = L e ; OL = L e : 1 1+s s 2 2+s s 3 3+s s s=0 s=0 s=0 Proposition 5.2. The generating function of Leonardo’s octonions for non-positive integer index, is expressed by: OL + OL x + (2OL + OL )x 0 1 0 2 g(OL ; x) = ; 3 2 x 2x + 1 with the respective initial values: 7 7 7 X X X OL = L e ; OL = L e and OL = L e : 2 2+s s 1 2+s s 0 s s s=0 s=0 s=0 Proof. Let us consider the function n 2 n g(OL ) = OL x = OL + OL x + OL x + : : : + OL x + : : : : n;x n 0 1 2 n n=0 2 3 Multiplying the function by 2x and x , we have: 2 2 3 4 n 2x g(OL ; x) = 2OL x + 2OL x + 2OL x + : : : + 2OL x + : : : ; n 0 1 2 n2 3 3 4 5 n x g(OL ; x) = OL x + OL x + OL x + : : : + OL x + : : : : n 0 1 2 n3 130 Renata Passos Machado Vieira et al. Subtracting the previous equalities and after some calculations, we obtain: 3 2 2 (x 2x + 1)g(OL ; x) = OL + OL x + (2OL + OL )x ; n 0 1 0 2 OL + OL x + (2OL + OL )x 0 1 0 2 g(OL ; x) = : 3 2 x 2x + 1 P P 7 7 Note that OL = L e , OL = L e and OL = 2 2+s s 1 2+s s 0 s=0 s=0 L e . s s s=0 Proposition 5.3. For n > 0 and n 2 N, the generator matrix of Leonardo’s octonions, with non-positive integer index, is given by: 2 3 2 3 OLF OF OF 2 0 0 0 1 n+2 6 7 OLF 4 5 3 1 1 1 0 2 OF OF 4 1 0 5 n+1 OLF 0 1 0 2 OF OF 0 1 2 3 OLF OF OF 2 0 n+2 6OLF 7 L L L OF OF n+2 n+1 n 4 1 0 5 n+1 OLF OF OF 0 1 OL OL OL = ; n+2 n+1 n where OLF = OL + OF . n n n Proof. Similar to the demonstration performed in Proposition 4.4, this property can be validated. 6. Leonardo’s Gaussians and octonions properties Next, some properties inherent to Leonardo’s Gaussians and octonions are studied. Proposition 6.1. The sum of the first n numbers of Leonardo’s Gaussians is given by: n n3 X X GL = 2GL + 2GL (2 + 4i) + GL : m n2 n1 s m=3 s=2 The generalization of Gaussians and Leonardo’s octonions 131 Proof. Using the recurrence relation of Leonardo’s Gaussians with n 2 N, we have: (6.1) GL = 2GL GL : n n1 n3 Thus, evaluating the relation given in (6.1) in values of n  3, we get: GL = 2GL GL ; 3 2 0 GL = 2GL GL ; 4 3 1 GL = 2GL GL ; 5 4 2 GL = 2GL GL ; 6 5 3 GL = 2GL GL ; 7 6 4 GL = 2GL GL ; n2 n3 n5 GL = 2GL GL ; n1 n2 n4 GL = 2GL GL : n n1 n3 Through successive cancellations, we obtain: GL = GL GL + GL GL + GL m 2 0 3 1 4 m=3 + + GL + 2GL + 2GL n3 n2 n1 n3 = 2GL + 2GL (GL + GL ) + GL : n2 n1 0 1 s s=2 Proposition 6.2. The sum of the numbers with even indexes of Leonardo’s Gaussians is given by: n n2 X X GL = 2GL GL + GL : 2m 2n1 1 2s+1 m=3 s=1 Proof. Using the recurrence relation of Leonardo’s Gaussians with n 2 N, we have: GL = 2GL GL : n n1 n3 132 Renata Passos Machado Vieira et al. Thus, evaluating the recurrence relation in values of n  3, we get: GL = 2GL GL ; 4 3 1 GL = 2GL GL ; 6 5 3 GL = 2GL GL ; 8 7 5 GL = 2GL GL ; 2n2 2n3 2n5 GL = 2GL GL : 2n 2n1 2n3 Through successive cancellations, we obtain: GL = GL GL + GL + + GL + 2GL 2m 3 1 5 2n3 2n1 m=2 n2 = 2GL GL + GL : 2n1 1 2s+1 s=1 Proposition 6.3. The sum of the odd index numbers of Leonardo’s Gaus- sians is given by: n n3 X X GL = 2GL GL + GL : 2m1 2n2 0 2s m=2 s=0 Proof. Using the recurrence relation of Leonardo’s Gaussians with n 2 N, we have: GL = 2GL GL : n n1 n3 Thus, evaluating the recurrence relation in values of n  3, we get: GL = 2GL GL ; 3 2 0 GL = 2GL GL ; 5 4 2 GL = 2GL GL ; 7 6 4 GL = 2GL GL ; 2n3 2n4 2n6 GL = 2GL GL : 2n1 2n2 2n4 The generalization of Gaussians and Leonardo’s octonions 133 Through successive cancellations, we obtain: GL = GL GL + GL + + GL + 2GL 2m1 2 0 4 2n4 2n2 m=2 n3 = 2GL GL + GL : 2n2 0 2s s=0 Proposition 6.4. The sum of the first n numbers of Leonardo’s octonions is given by: n 7 n3 X X X OL = 2OL + 2OL (L + L )e + OL : m n2 n1 s s+1 s s m=3 s=0 s=2 Proof. Using the recurrence relation of Leonardo’s octonions with n 2 N, we have: (6.2) OL = 2OL OL : n n1 n3 Thus, evaluating the relation given in (6.2) in values of n  3, we get: OL = 2OL OL ; 3 2 0 OL = 2OL OL ; 4 3 1 OL = 2OL OL ; 5 4 2 OL = 2OL OL ; 6 5 3 OL = 2OL OL ; 7 6 4 OL = 2OL OL ; n2 n3 n5 OL = 2OL OL ; n1 n2 n4 OL = 2OL OL : n n1 n3 Through successive cancellations, we obtain: OL = OL OL + OL OL + OL + + 2OL + 2OL m 2 0 3 1 4 n2 n1 m=3 n3 = 2OL + 2OL (OL + OL ) + OL : n2 n1 0 1 s s=2 134 Renata Passos Machado Vieira et al. Considering the initial values through Definition 4.1, it is concluded that: 7 n3 X X 2OL + 2OL (L + L )e + OL : n2 n1 s+0 s+1 s s s=0 s=2 Proposition 6.5. The sum of the numbers with even indexes of Leonardo’s octonions is given by: n 7 n2 X X X OL = 2OL L e + OL : 2m 2n1 s+1 s 2s+1 m=2 s=0 s=1 Proof. Using the recurrence relation of Leonardo’s octonsion with n 2 N, we have: OL = 2OL OL : n n1 n3 Thus, evaluating the recurrence relation in values of n  3, we get: OL = 2OL OL ; 4 3 1 OL = 2OL OL ; 6 5 3 OL = 2OL OL ; 8 7 5 OL = 2OL OL ; 2n2 2n3 2n5 OL = 2OL OL : 2n 2n1 2n3 Through successive cancellations, we obtain: OL = OL OL + OL + + OL + 2OL 2m 3 1 5 2n3 2n1 m=2 2n3 = 2OL OL + OL : 2n1 1 s s=3 Considering the initial values through Definition 4.1, it follows that: 7 n2 X X 2OL L e + OL : 2n1 s+1 s 2s+1 s=0 s=1 The generalization of Gaussians and Leonardo’s octonions 135 Proposition 6.6. The sum of the numbers with odd indices of Leonardo’s octonions is given by: n 7 n3 X X X OL = 2OL L e + OL : 2m1 2n2 s+0 s 2s+2 m=2 s=0 s=0 Proof. Using the recurrence relation of Leonardo’s octonions with n 2 N, we have: OL = 2OL OL : n n1 n3 Thus, evaluating the recurrence relation, in values of n  3, we get: OL = 2OL OL ; 3 2 0 OL = 2OL OL ; 5 4 2 OL = 2OL OL ; 7 6 4 OL = 2OL OL ; 2n3 2n4 2n6 OL = 2: 2n1 Through successive cancellations, we obtain: OL = OL OL + OL + + OL + 2OL 2m1 2 0 4 2n4 2n2 m=2 n3 = 2OL OL + OL : 2n2 0 2s+1 s=0 Considering the initial values through Definition 4.1, it follows that: 7 2n4 X X 2OL L e + OL : 2n2 s+0 s s s=0 s=2 136 Renata Passos Machado Vieira et al. 7. Conclusion This work presents a discussion about the evolutionary process of Leonardo’s sequence. When complexifying this sequence, it is possible to present the dimensional growth of the sequence from the insertion of the imag- inary unit i, thus presenting Leonardo’s Gaussians. And yet, it was possible to approach the terms of Leonardo’s sequence in eight dimensions, obtaining Leonardo’s octonions. Moreover, the generating functions, Binet’s formula, matrix forms, gener- alizations and properties linked to these numbers were also presented. Finally, this article makes it possible to contribute to the mathematical field and pro- vides mathematical researchers with knowledge about Leonardo’s sequence and its evolutionary process. References [1] F.R.V. Alves and R.P.M. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab, Int. Elect. J. Math. Ed. 15 (2020), no. 2, Article No. em0575, 9 pp. [2] F.R.V. Alves, R.P.M. Vieira, and P.M.M.C. Catarino, Visualizing the Newtons fractal from the recurring linear sequence with Google Colab: An example of Brazil X Portugal research, Int. Elect. J. Math. Ed. 15 (2020), no. 3, Article No. em0594, 19 pp. [3] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. (N.S.) 89 (2020), no. 1, 75–86. [4] C.J. Harman, Complex Fibonacci numbers, Fibonacci Quart. 19 (1981), no. 1, 82–86. [5] A. Karataş and S. Halici, Horadam octonions, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 25 (2017), no. 3, 97–106. [6] O. Keçilioğlu and I. Akkus, The Fibonacci octonions, Adv. Appl. Clifford Algebr. 25 (2015), no. 1, 151–158. [7] A.G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Dis- crete Math. 25 (2019), no. 3, 97–101. [8] R.P.M. Vieira, F.R.V. Alves, and P.M.M.C. Catarino, Relações bidimensionais e iden- tidades da sequência de Leonardo, Revista Sergipana de Matemática e Educação Matemática 4 (2019), no. 2, 156–173. [9] R.P.M. Vieira, F.R.V. Alves, and P.M.M.C. Catarino, Uma extensão dos octônios de Padovan para inteiros não positivos, C.Q.D. – Revista Eletrônica Paulista de Matemática 19 (2020), Edição Dezembro, 9–16. [10] R.P.M. Vieira, M.C. dos S. Mangueira, F.R.V. Alves, and P.M.M.C. Catarino, A forma matricial dos números de Leonardo, Ci. e Nat. 42 (2020), 40 yrs. – Anniv. Ed., Article No. e100, 6 pp. The generalization of Gaussians and Leonardo’s octonions 137 Renata Passos Machado Vieira Federal University of Ceará Fortaleza-CE Brasil e-mail: re.passosm@gmail.com Milena Carolina dos Santos Mangueira Federal Institute of Ceará Fortaleza-CE Brasil e-mail: milenacarolina24@gmail.com Francisco Régis Vieira Alves Federal Institute of Ceará Fortaleza-CE Brasil e-mail: fregis@ifce.edu.br Paula Maria Machado Cruz Catarino University of Trás-os-Montes and Alto Douro Vila Real Portugal e-mail: pcatarino23@gmail.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Mathematicae Silesianae de Gruyter

The Generalization of Gaussians and Leonardo’s Octonions

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Annales Mathematicae Silesianae 37 (2023), no. 1, 117–137 DOI: 10.2478/amsil-2023-0004 THE GENERALIZATION OF GAUSSIANS AND LEONARDO’S OCTONIONS Renata Passos Machado Vieira , Milena Carolina dos Santos Mangueira, Francisco Régis Vieira Alves, Paula Maria Machado Cruz Catarino Abstract. In order to explore the Leonardo sequence, the process of complex- ification of this sequence is carried out in this work. With this, the Gaussian and octonion numbers of the Leonardo sequence are presented. Also, the re- currence, generating function, Binet’s formula, and matrix form of Leonardo’s Gaussian and octonion numbers are defined. The development of the Gauss- ian numbers is performed from the insertion of the imaginary component i in the one-dimensional recurrence of the sequence. Regarding the octonions, the terms of the Leonardo sequence are presented in eight dimensions. Further- more, the generalizations and inherent properties of Leonardo’s Gaussians and octonions are presented. 1. Introduction Historically, the origin of the Leonardo sequence reports that this sequence was possibly created by Leonardo of Pisa (1180-1250) [2]. This fact is due to its great similarity with the Fibonacci sequence and also because the sequence Received: 11.01.2022. Accepted: 06.02.2023. Published online: 27.02.2023. (2020) Mathematics Subject Classification: 11B37, 11B39. Key words and phrases: matrix form, Binet’s formula, Gaussian, octonions, Leonardo sequence. The part of research development in Brazil had the financial support of the National Council for Scientific and Technological Development – CNPq and the Ceará Foundation for Support to Scientific and Technological Development (Funcap). The research development aspect in Portugal is financed by National Funds through FCT – Fundação para a Ciência e Tecnologia. I.P, within the scope of the UID/CED/00194/2020 project. 2023 The Author(s). This is an Open Access article distributed under the terms of the Creative Commons Attribution License CC BY (http://creativecommons.org/licenses/by/4.0/). 118 Renata Passos Machado Vieira et al. has the same name as the Italian mathematician. It is possible to find the mathematical evolution of this sequence in the works [3, 1, 7, 10, 8]. The Leonardo sequence satisfies the following recurrence relation: L = L + L + 1; n  2; L = L = 1: n n1 n2 0 1 On the other hand, for n + 1 we can rewrite this recurrence relation as L = L + L + 1. With this, we can add the term L , resulting in n+1 n n1 n+1 another recurrence relation. Hence, we have: L L = L + L + 1 L L 1; n n+1 n1 n2 n n1 i.e., L = 2L L ; n+1 n n2 where L = L = 1 are initial conditions. 0 1 In 1981 Harman ([4]) introduced the Gaussian numbers denoted by z = a + bi with a, b 2 Z and i = 1, which will be associated with the Leonardo sequence, improving the process of mathematical complexification of this se- quence. Regarding octonion numbers, according to Vieira, Alves, and Catarino (2020) [9], it can be said that in the work of Karatas and Halici (2017) [5] the algebra in sixteen dimensions was studied and Horadam octonions by Ho- radam sequence which is a generalization of second order recurrence relations were defined. Octonions were defined as the R numerical field, writing as ([6, 5]): 0 00 p = p + p e; 0 00 2 2 2 where p ; p 2 H = fa + a iii + a jjj + a kkk j iii = jjj = kkk = 1;ij ij ijkkk = 1; 0 1 2 3 a ; a ; a ; a 2 Rg. 0 1 2 3 For the operation of adding and multiplying between two octonions, 0 00 0 00 p = p + p e and, q = q + q e: 0 0 00 00 p + q = (p + q ) + (p + q )e; 0 0 00 00 0 00 00 0 pq = (p q q p ) + (q p + p q )e; 0 00 0 00 where q and q are the conjugates of the quaternions q and q , respectively. Therefore, O is the algebra of the octonions, on a natural basis in the space over R formed by the elements: e = 1, e = iii, e = jjj, e = kkk, e = e, e = iiie, 0 1 2 3 4 5 e = jjje, e = kkke and the multiplication of these numbers is shown in the 6 7 Table 1. The generalization of Gaussians and Leonardo’s octonions 119 Table 1. Multiplication of the octonions of O. Source: [6] 1 e e e e e e e 1 2 3 4 5 6 7 1 1 e e e e e e e 1 2 3 4 5 6 7 e e -1 e -e e -e -e e 1 1 3 2 5 4 7 6 e e -e -1 e e e -e -e 2 2 3 1 6 7 4 5 e e e -e -1 e -e e -e 3 3 2 1 7 6 5 4 e e -e -e -e -1 e e e 4 4 5 6 7 1 2 3 e e e -e e -e -1 -e e 5 5 4 7 6 1 3 2 e e e e -e -e e -1 -e 6 6 7 4 5 2 3 1 e e -e e e -e -e e -1 7 7 6 5 4 3 2 1 Thus, the following notation is used for octonions: p = p e ; s s p=0 where p is the real coefficient, with p 2 O, in format p = Re(p) + Im(p), where Re(p) = p represents the real part and, Im(p) = p e represents 0 s s s=1 the imaginary part. Therefore, in this work, it is intended to continue the mathematical evo- lution of this sequence, presenting the Gaussian and octonion numbers of Leonardo’s sequence. 2. Leonardo’s Gaussians Henceforward, Leonardo’s Gaussian numbers will be introduced, beginning complex studies around this sequence, with the insertion of an imaginary unit. Thus, their respective mathematical aspects are portrayed. Definition 2.1. For n > 0, Leonardo’s Gaussians are defined by: GL = L + iL ; n n n+1 where L = L = 1. In particular, GL = 1 + i; GL = 1 + 3i. 0 1 0 1 From the previous definition, it is easy to see that for all n > 3 and n 2 N, the recurrence formula of Leonardo’s Gaussian is given by: GL = 2GL GL ; n n1 n3 where GL = 1 + i and GL = 1 + 3i. 0 1 120 Renata Passos Machado Vieira et al. Definition 2.2. For n > 0, Fibonacci’s Gaussians are defined by: GF = F + iF ; n n n+1 where F = 0; F = 1. In particular, GF = i and GF = 1 + i. 0 1 0 1 From the previous definition, it is easy to see that for all n > 2 and n 2 N, the recurrence formula of Fibonacci’s Gaussian is given by: GF = GF + GF ; n n1 n2 where GF = i and GF = 1 + i. 0 1 Definition 2.3. Leonardo and Fibonacci’s Gaussian recurrence formula is given by: GLF = GL + GF ; n n n where n 2 Z. Theorem 2.4. The generating function of Leonardo’s Gaussians is given by: 1 + i + (1 i)(x + x ) g(GL ; x) = : (1 2x x ) Proof. Let us consider the function 2 n g(GL ; x) = GL + GL x + GL x + : : : + GL x + : : : : n 0 1 2 n Multiplying this function by 2x and x , we get 2 3 n 2xg(GL ; x) = 2GL x + 2GL x + 2GL x + : : : + 2GL x + : : : ; n 0 1 2 n1 3 3 4 5 n x g(GL ; x) = GL x + GL x + GL x + : : : + GL x + : : : : n 0 1 2 n3 Subtracting the previous equalities and after some calculations, we obtain: 3 2 (1 2x x )g(GL ; x) = GL + (GL 2GL )x + (GL 2GL )x ; n 0 1 0 2 1 3 2 (1 2x x )g(GL ; x) = 1 + i (1 i)x + (1 i)x ; 3 2 (1 2x x )g(GL ; x) = 1 + i + (1 i)(x + x ); 1 + i + (1 i)(x + x ) g(GL ; x) = : (1 2x x ) The generalization of Gaussians and Leonardo’s octonions 121 Theorem 2.5. The Binet formula of Leonardo’s Gaussians, with n 2 Z, is: n n n GL = A (1 + ir )r + B (1 + ir )r + C (1 + ir )r ; n g 1 g 2 g 3 1 2 3 3 2 where r ; r ; r are the roots of the characteristic polynomial r 2r + 1 = 0, 1 2 3 (r 1)(r 1) (r 1)(r 1) 2 3 1 3 A = ; B = ; g g (r r )(r r ) (r r )(r r ) 1 2 1 3 2 1 2 3 (r 1)(r 1) 1 2 C = : (r r )(r r ) 3 1 3 2 n n n Proof. Through the Binet’s formula GL = r + r + r and the 1 2 3 recurrence of Leonardo’s Gaussians GL = L +iL , with the initial values n n n+1 GL = 1 + i, GL = 1 + 3i and GL = 3 + 5i, it is possible to obtain the 0 1 2 following system of equations: + + = 1 + i; r + r + r = 1 + 3i; 1 2 3 2 2 2 r + r + r = 3 + 5i: 1 2 3 Solving the system, we get: (3 + 5i) + (r r )(1 + 3i) + r r (1 + i) 2 3 2 3 = ; r r r r r + r r 1 2 1 3 2 3 (3 + 5i) + (r r )(1 + 3i) + r r (1 + i) 1 3 1 3 = ; r r r r r + r r 2 3 1 2 1 3 (3 + 5i) + (r r )(1 + 3i) + r r (1 + i) 1 2 1 2 = : r + r r r r r r 1 2 1 3 2 3 Through Girard’s relations: r r r = 1; r +r +r = 2 and r r +r r + 1 2 3 1 2 3 1 2 2 3 r r = 0, it is easy to see that: 1 3 (r r r r + 1) (r 1)(r 1) 2 2 2 3 2 3 = (1 + ir )= (1 + ir ) =A (1 + ir ); 1 1 g 1 (r r )(r r ) (r r )(r r ) 1 2 1 3 1 2 1 3 (r r r r + 1) (r 1)(r 1) 1 3 1 3 1 3 = (1 + ir )= (1 + ir ) =B (1 + ir ); 2 2 g 2 (r r )(r r ) (r r )(r r ) 2 1 2 3 2 1 2 3 (r r r r + 1) (r 1)(r 1) 1 2 1 2 1 2 = (1 + ir )= (1 + ir ) =C (1 + ir ): 3 3 g 3 (r r )(r r ) (r r )(r r ) 3 1 3 2 3 1 3 2 122 Renata Passos Machado Vieira et al. Based on the work of Vieira, Mangueira, Alves and Catarino ([10]), one can establish the matrix form of Leonardo’s sequence in the complex form. Proposition 2.6. For n > 2 and n 2 N, the matrix form of Leonardo’s Gaussians is given by: 2 3 2 3 GLF GF GF 2 0 2 1 0 L n+2 6GLF 7 4 5 3 1 1 0 0 1 4 GF GF 5 1 0 n+1 GLF 1 0 0 2 GF GF 0 1 2 3 GLF GF GF 2 0 n+2 6 7 GLF = L L L GF GF n+2 n+1 n 4 5 1 0 n+1 GLF GF GF 0 1 GL GL GL = ; n+2 n+1 n where GLF = GL + GF , for n < 0. n n n Proof. By the principle of finite induction, we have for n = 2: 2 3 2 3 GLF 2 2 GF GF 2 0 2 1 0 6GLF 7 4 5 3 1 1 0 0 1 4 GF GF 5 1 0 GLF 1 0 0 GF GF 0 1 2 3 GLF GF GF 2 0 GLF 4 5 9 5 3 = GF GF 1 0 GLF GF GF 0 1 = 9GF 3GF +GLF 9GF +5GF +GLF GLF +5GF +3GF 2 0 2 0 1 2 2 0 1 GL GL GL = : 4 3 2 So, assume it is true for any n = k, with k 2 N: 2 3 2 3 GLF k 2 GF GF 2 0 2 1 0 L k+2 6 7 GLF 4 5 3 1 1 0 0 1 GL GL GL GF GF = : 4 1 0 5 k+2 k+1 k k+1 1 0 0 GLF GF GF 0 1 k The generalization of Gaussians and Leonardo’s octonions 123 Let us show that it is still valid for n = k + 1: 2 3 2 3 GLF k+1 GF GF 2 0 2 1 0 L k+3 6GLF 7 4 5 3 1 1 0 0 1 OF GF 4 1 0 5 k+2 GLF 1 0 0 2 GF GF 0 1 k+1 2 3 2 3 GLF GF GF 2 0 2 1 0 L k+3 6GLF 7 4 5 = L L L 0 0 1 GF GF 4 5 k+2 k+1 k 1 0 k+2 GLF 1 0 0 2 GF GF 0 1 k+1 2 3 GLF GF GF 2 0 k+3 6 7 GLF = L L L GF GF 4 1 0 5 k+3 k+2 k+1 k+2 GLF GF GF 0 1 k+1 = [L GF L GF + GLF L GF + L GF + GLF k+3 2 k+1 0 2 k+3 0 k+2 1 2 L GF + L GF + GLF ] k+2 0 k+1 1 2 = GL GL GL : k+3 k+2 k+1 3. The generalization of Leonardo’s Gaussians Next, the behavior of terms with non-positive integer indices of Leonardo’s Gaussians will be analyzed. Definition 3.1. For all n > 0 and n 2 N, Leonardo’s Gaussians, for non-positive integer index, are defined by the equation: GL = L e : n n+s s s=0 From the previous definition, it is easy to see that for all n > 0 and n 2 N, the recurrence formula of Leonardo’s Gaussians for non-positive integer index, is given by: GL = 2GL GL ; n n+2 n+3 where GL = 1 + i; GL = 1 + 3i and GL = 4 + 6i. 0 1 2 124 Renata Passos Machado Vieira et al. Proposition 3.2. The generating function of Leonardo’s Gaussians for non-positive integer index, is expressed by: 1 + i + (1 + i)x + (1 3i)x g(GL ; x) = : 3 2 x 2x + 1 Proof. Let us consider the function n 2 n g(GL ) = GL x = GL + GL x + GL x + : : : + GL x + : : : : n;x n 0 1 2 n n=0 2 3 Multiplying this function by 2x and x , we have: 2 2 3 4 n 2x g(GL ; x) = 2GL x + 2GL x + 2GL x + : : : + 2GL x + : : : ; n 0 1 2 n2 3 3 4 5 n x g(GL ; x) = GL x + GL x + GL x + : : : + GL x + : : : : n 0 1 2 n3 Subtracting the previous equalities and after some calculations, we obtain: 3 2 2 (x 2x + 1)g(GL ; x) = GL + GL x + (2GL + GL )x ; n 0 1 0 2 3 2 2 (x 2x + 1)g(GL ; x) = 1 + i + (1 + i)x + (1 3i)x ; 1 + i + (1 + i)x + (1 3i)x g(GL ; x) = : 3 2 x 2x + 1 Proposition 3.3. For n>0 and n2N, the generating matrix of Leonardo’s Gaussians, with a non-positive integer index, is given by: 2 3 2 3 GLF GF GF 2 0 0 0 1 n+2 6 7 GLF 4 5 3 1 1 1 0 2 GF GF 4 1 0 5 n+1 GLF 0 1 0 2 GF GF 0 1 2 3 GLF GF GF 2 0 n+2 6GLF 7 L L L GF GF n+2 n+1 n 4 1 0 5 n+1 GLF GF GF 0 1 GL GL GL = ; n+2 n+1 n where GLF = GL + GF , GF = i 1, GF = 1 and GF = i. n n n 2 1 0 Proof. Similarly to the demonstration performed in Proposition 2.6, this property can be validated.  The generalization of Gaussians and Leonardo’s octonions 125 4. Leonardo’s octonions In this section, Leonardo’s octonions will be studied, addressing their re- spective mathematical properties. Definition 4.1. For n > 0, Leonardo’s octonions are defined by: OL = L e ; n n+s s s=0 P P P 7 7 7 where OL = L e , OL = L e , OL = L e : 0 s s 1 1+s s 2 2+s s s=0 s=0 s=0 From the previous definition, it is easy to see that for all n > 3 and n 2 N, the recurrence formula of Leonardo’s octonions is given by: OL = 2OL OL ; n n1 n3 P P P 7 7 7 where OL = L e , OL = L e , OL = L e . 0 s s 1 1+s s 2 2+s s s=0 s=0 s=0 Theorem 4.2. The generating function of Leonardo’s octonions, OL , is given by: g(OL ; x) = (L + L x + L x )e : n s s2 s1 s (1 2x x ) s=0 Proof. Let us consider the function 2 n g(OL ; x) = OL + OL x + OL x + : : : + OL x + : : : : n 0 1 2 n Multiplying the function by 2x and x , we get: 2 3 n 2xg(OL ; x) = 2OL x + 2OL x + 2OL x + : : : + 2OL x + : : : ; n 0 1 2 n1 3 3 4 5 n x g(OL ; x) = OL x + OL x + OL x + : : : + OL x + : : : : n 0 1 2 n3 Subtracting the previous equalities and after some calculations, we obtain: 3 2 (1 2x x )g(OL ; x) = OL + (OL 2OL )x + (OL 2OL )x ; n 0 1 0 2 1 g(OL ; x) = (L + L x + L x )e : n s s2 s1 s (1 2x x ) s=0 126 Renata Passos Machado Vieira et al. Theorem 4.3. Binet’s formula for Leonardo’s octonions, with n 2 Z, is given by: n n n OL = r + r + r ; 1 2 3 3 2 where r ; r and r are the roots of the characteristic polynomial r 2r +1=0, 1 2 3 (r 1)(r 1) (r 1)(r 1) (r 1)(r 1) 2 3 1 3 1 2 A = ; B = ; C = ; l l l (r r )(r r ) (r x )(r r ) (r r )(r r ) 1 2 1 3 2 1 2 3 3 1 3 2 7 7 7 X X X s s s = r e ; = r e ; = r e ; ol s ol s ol s 1 2 3 s=0 s=0 s=0 = A ; = B ; = C : l ol l ol l ol n n n Proof. Through the Binet’s formula oL = r + r + r and the 1 2 3 recurrence of Leonardo’s octonions OL = L e , with the initial val- n n+s s s=0 P P P 7 7 7 ues OL = L e , OL = L e and OL = L e , it is 0 s s 1 s+1 s 2 s+2 s s=0 s=0 s=0 possible to obtain the following system of equations: + + = L e ; > s s s=0 r + r + r = L e ; 1 2 3 s+1 s s=0 > 7 > X 2 2 2 > r + r + r = L e : s+2 s : 1 2 3 s=0 Solving this system, we have: 7 7 7 X X X L e + (r r ) L e + r r L e s+2 s 2 3 s+1 s 2 3 s s s=0 s=0 s=0 = ; r r r r r + r r 1 2 1 3 2 3 7 7 7 X X X L e + (r r ) L e + r r L e s+2 s 1 3 s+1 s 1 3 s s s=0 s=0 s=0 = ; r r r r r + r r 2 3 1 2 1 3 7 7 7 X X X L e + (r r ) L e + r r L e s+2 s 1 2 s+1 s 1 2 s s s=0 s=0 s=0 = : r + r r r r r r 1 2 1 3 2 3 3 The generalization of Gaussians and Leonardo’s octonions 127 Through Girard’s relations: r x r = 1; r + r + r = 2 and r r + r r + 1 2 3 1 2 3 1 2 2 3 r r = 0, it is easy to see that: 1 3 7 7 7 X X X (r r r r + 1) (r 1)(r 1) 2 2 2 3 2 3 s s s = r e = r e = A r e ; s s l s 1 1 1 (r r )(r r ) (r r )(r r ) 1 2 1 3 1 2 1 3 s=0 s=0 s=0 7 7 7 X X X (r r r r + 1) (r 1)(r 1) 1 3 1 3 1 3 s s s = r e = r e = B r e ; s s l s 2 2 2 (r r )(r r ) (r r )(r r ) 2 1 2 3 2 1 2 3 s=0 s=0 s=0 7 7 7 X X X (r r r r + 1) (r 1)(r 1) 1 2 1 2 1 2 s s s = r e = r e = C r e : s s l s 3 3 3 (r r )(r r ) (r r )(r r ) 3 1 3 2 3 1 3 2 s=0 s=0 s=0 P P P 7 7 7 s s s Defining = r e , = r e and = r e , it is easy to ol s ol s ol s 1 2 3 s=0 s=0 s=0 see that: = A ; = B ; = C : l ol l ol l ol The matrix form of Leonardo’s octonions is based on the work of Vieira, Mangueira, Alves and Catarino ([10]), in which we found a development on the matrix form of the one-dimensional Leonardo sequence. Proposition 4.4. For n > 2 and n 2 N, the matrix form of Leonardo’s octonions is given by: 2 3 2 3 OLF OF OF 2 0 2 1 0 L n+2 6OLF 7 4 5 3 1 1 0 0 1 4 OF OF 5 1 0 n+1 OLF 1 0 0 2 OF OF 0 1 2 3 OLF OF OF 2 0 n+2 6 7 OLF = L L L OF OF n+2 n+1 n 4 5 1 0 n+1 OLF OF OF 0 1 OL OL OL = ; n+2 n+1 n where OLF = OL + OF . n n n 128 Renata Passos Machado Vieira et al. Proof. By the principle of finite induction, we have for n = 2: 2 3 2 3 OLF 2 2 OF OF 2 0 2 1 0 6 7 OLF 4 5 3 1 1 0 0 1 OF OF 4 5 1 0 OLF 1 0 0 OF OF 0 1 2 3 OLF OF OF 2 0 OLF 4 5 9 5 3 = OF OF 1 0 OLF OF OF 0 1 = 9OF 3OF +OLF 9OF +5OF +OLF OLF +5OF +3OF 2 0 2 0 1 2 2 0 1 OL OL OL = : 4 3 2 So, assume it is true for any n = k, with k 2 N: 2 3 2 3 OLF k 2 OF OF 2 0 2 1 0 L k+2 6 7 OLF 4 5 3 1 1 0 0 1 OL OL OL OF OF = : 4 1 0 5 k+2 k+1 k k+1 1 0 0 OLF OF OF 0 1 Finally, the validity for n = k + 1 is verified: 2 3 2 3 OLF k+1 OF OF 2 0 2 1 0 k+3 6OLF 7 4 5 3 1 1 0 0 1 OF OF 4 1 0 5 k+2 OLF 1 0 0 2 OF OF 0 1 k+1 2 3 2 3 OLF OF OF 2 0 2 1 0 L k+3 6OLF 7 4 5 L L L 0 0 1 OF OF k+2 k+1 k 4 1 0 5 k+2 OLF 1 0 0 2 OF OF 0 1 k+1 2 3 OLF OF OF 2 0 k+3 6OLF 7 = L L L OF OF 4 5 k+3 k+2 k+1 1 0 k+2 OLF OF OF 0 1 k+1 = [L OF L OF +OLF L OF +L OF +OLF k+3 2 k+1 0 2 k+3 0 k+2 1 2 L OF +L OF +OLF ] k+2 0 k+1 1 2 = OL OL OL : k+3 k+2 k+1 The generalization of Gaussians and Leonardo’s octonions 129 5. The generalization of Leonardo’s octonions Next, the behavior of terms with non-positive integer indices of Leonardo’s octonions will be analyzed. Definition 5.1. For all n > 0 and n 2 N, Leonardo’s octonions, for non-positive integer index, are defined by the equation: OL = L e : n n+s s s=0 From the previous definition, it is easy to see that for all n > 0 and n 2 N, the recurrence formula of Leonardo’s octonions for non-positive integer index, is given by: OL = 2OL OL ; n n+2 n+3 P P P 7 7 7 where OL = L e ; OL = L e ; OL = L e : 1 1+s s 2 2+s s 3 3+s s s=0 s=0 s=0 Proposition 5.2. The generating function of Leonardo’s octonions for non-positive integer index, is expressed by: OL + OL x + (2OL + OL )x 0 1 0 2 g(OL ; x) = ; 3 2 x 2x + 1 with the respective initial values: 7 7 7 X X X OL = L e ; OL = L e and OL = L e : 2 2+s s 1 2+s s 0 s s s=0 s=0 s=0 Proof. Let us consider the function n 2 n g(OL ) = OL x = OL + OL x + OL x + : : : + OL x + : : : : n;x n 0 1 2 n n=0 2 3 Multiplying the function by 2x and x , we have: 2 2 3 4 n 2x g(OL ; x) = 2OL x + 2OL x + 2OL x + : : : + 2OL x + : : : ; n 0 1 2 n2 3 3 4 5 n x g(OL ; x) = OL x + OL x + OL x + : : : + OL x + : : : : n 0 1 2 n3 130 Renata Passos Machado Vieira et al. Subtracting the previous equalities and after some calculations, we obtain: 3 2 2 (x 2x + 1)g(OL ; x) = OL + OL x + (2OL + OL )x ; n 0 1 0 2 OL + OL x + (2OL + OL )x 0 1 0 2 g(OL ; x) = : 3 2 x 2x + 1 P P 7 7 Note that OL = L e , OL = L e and OL = 2 2+s s 1 2+s s 0 s=0 s=0 L e . s s s=0 Proposition 5.3. For n > 0 and n 2 N, the generator matrix of Leonardo’s octonions, with non-positive integer index, is given by: 2 3 2 3 OLF OF OF 2 0 0 0 1 n+2 6 7 OLF 4 5 3 1 1 1 0 2 OF OF 4 1 0 5 n+1 OLF 0 1 0 2 OF OF 0 1 2 3 OLF OF OF 2 0 n+2 6OLF 7 L L L OF OF n+2 n+1 n 4 1 0 5 n+1 OLF OF OF 0 1 OL OL OL = ; n+2 n+1 n where OLF = OL + OF . n n n Proof. Similar to the demonstration performed in Proposition 4.4, this property can be validated. 6. Leonardo’s Gaussians and octonions properties Next, some properties inherent to Leonardo’s Gaussians and octonions are studied. Proposition 6.1. The sum of the first n numbers of Leonardo’s Gaussians is given by: n n3 X X GL = 2GL + 2GL (2 + 4i) + GL : m n2 n1 s m=3 s=2 The generalization of Gaussians and Leonardo’s octonions 131 Proof. Using the recurrence relation of Leonardo’s Gaussians with n 2 N, we have: (6.1) GL = 2GL GL : n n1 n3 Thus, evaluating the relation given in (6.1) in values of n  3, we get: GL = 2GL GL ; 3 2 0 GL = 2GL GL ; 4 3 1 GL = 2GL GL ; 5 4 2 GL = 2GL GL ; 6 5 3 GL = 2GL GL ; 7 6 4 GL = 2GL GL ; n2 n3 n5 GL = 2GL GL ; n1 n2 n4 GL = 2GL GL : n n1 n3 Through successive cancellations, we obtain: GL = GL GL + GL GL + GL m 2 0 3 1 4 m=3 + + GL + 2GL + 2GL n3 n2 n1 n3 = 2GL + 2GL (GL + GL ) + GL : n2 n1 0 1 s s=2 Proposition 6.2. The sum of the numbers with even indexes of Leonardo’s Gaussians is given by: n n2 X X GL = 2GL GL + GL : 2m 2n1 1 2s+1 m=3 s=1 Proof. Using the recurrence relation of Leonardo’s Gaussians with n 2 N, we have: GL = 2GL GL : n n1 n3 132 Renata Passos Machado Vieira et al. Thus, evaluating the recurrence relation in values of n  3, we get: GL = 2GL GL ; 4 3 1 GL = 2GL GL ; 6 5 3 GL = 2GL GL ; 8 7 5 GL = 2GL GL ; 2n2 2n3 2n5 GL = 2GL GL : 2n 2n1 2n3 Through successive cancellations, we obtain: GL = GL GL + GL + + GL + 2GL 2m 3 1 5 2n3 2n1 m=2 n2 = 2GL GL + GL : 2n1 1 2s+1 s=1 Proposition 6.3. The sum of the odd index numbers of Leonardo’s Gaus- sians is given by: n n3 X X GL = 2GL GL + GL : 2m1 2n2 0 2s m=2 s=0 Proof. Using the recurrence relation of Leonardo’s Gaussians with n 2 N, we have: GL = 2GL GL : n n1 n3 Thus, evaluating the recurrence relation in values of n  3, we get: GL = 2GL GL ; 3 2 0 GL = 2GL GL ; 5 4 2 GL = 2GL GL ; 7 6 4 GL = 2GL GL ; 2n3 2n4 2n6 GL = 2GL GL : 2n1 2n2 2n4 The generalization of Gaussians and Leonardo’s octonions 133 Through successive cancellations, we obtain: GL = GL GL + GL + + GL + 2GL 2m1 2 0 4 2n4 2n2 m=2 n3 = 2GL GL + GL : 2n2 0 2s s=0 Proposition 6.4. The sum of the first n numbers of Leonardo’s octonions is given by: n 7 n3 X X X OL = 2OL + 2OL (L + L )e + OL : m n2 n1 s s+1 s s m=3 s=0 s=2 Proof. Using the recurrence relation of Leonardo’s octonions with n 2 N, we have: (6.2) OL = 2OL OL : n n1 n3 Thus, evaluating the relation given in (6.2) in values of n  3, we get: OL = 2OL OL ; 3 2 0 OL = 2OL OL ; 4 3 1 OL = 2OL OL ; 5 4 2 OL = 2OL OL ; 6 5 3 OL = 2OL OL ; 7 6 4 OL = 2OL OL ; n2 n3 n5 OL = 2OL OL ; n1 n2 n4 OL = 2OL OL : n n1 n3 Through successive cancellations, we obtain: OL = OL OL + OL OL + OL + + 2OL + 2OL m 2 0 3 1 4 n2 n1 m=3 n3 = 2OL + 2OL (OL + OL ) + OL : n2 n1 0 1 s s=2 134 Renata Passos Machado Vieira et al. Considering the initial values through Definition 4.1, it is concluded that: 7 n3 X X 2OL + 2OL (L + L )e + OL : n2 n1 s+0 s+1 s s s=0 s=2 Proposition 6.5. The sum of the numbers with even indexes of Leonardo’s octonions is given by: n 7 n2 X X X OL = 2OL L e + OL : 2m 2n1 s+1 s 2s+1 m=2 s=0 s=1 Proof. Using the recurrence relation of Leonardo’s octonsion with n 2 N, we have: OL = 2OL OL : n n1 n3 Thus, evaluating the recurrence relation in values of n  3, we get: OL = 2OL OL ; 4 3 1 OL = 2OL OL ; 6 5 3 OL = 2OL OL ; 8 7 5 OL = 2OL OL ; 2n2 2n3 2n5 OL = 2OL OL : 2n 2n1 2n3 Through successive cancellations, we obtain: OL = OL OL + OL + + OL + 2OL 2m 3 1 5 2n3 2n1 m=2 2n3 = 2OL OL + OL : 2n1 1 s s=3 Considering the initial values through Definition 4.1, it follows that: 7 n2 X X 2OL L e + OL : 2n1 s+1 s 2s+1 s=0 s=1 The generalization of Gaussians and Leonardo’s octonions 135 Proposition 6.6. The sum of the numbers with odd indices of Leonardo’s octonions is given by: n 7 n3 X X X OL = 2OL L e + OL : 2m1 2n2 s+0 s 2s+2 m=2 s=0 s=0 Proof. Using the recurrence relation of Leonardo’s octonions with n 2 N, we have: OL = 2OL OL : n n1 n3 Thus, evaluating the recurrence relation, in values of n  3, we get: OL = 2OL OL ; 3 2 0 OL = 2OL OL ; 5 4 2 OL = 2OL OL ; 7 6 4 OL = 2OL OL ; 2n3 2n4 2n6 OL = 2: 2n1 Through successive cancellations, we obtain: OL = OL OL + OL + + OL + 2OL 2m1 2 0 4 2n4 2n2 m=2 n3 = 2OL OL + OL : 2n2 0 2s+1 s=0 Considering the initial values through Definition 4.1, it follows that: 7 2n4 X X 2OL L e + OL : 2n2 s+0 s s s=0 s=2 136 Renata Passos Machado Vieira et al. 7. Conclusion This work presents a discussion about the evolutionary process of Leonardo’s sequence. When complexifying this sequence, it is possible to present the dimensional growth of the sequence from the insertion of the imag- inary unit i, thus presenting Leonardo’s Gaussians. And yet, it was possible to approach the terms of Leonardo’s sequence in eight dimensions, obtaining Leonardo’s octonions. Moreover, the generating functions, Binet’s formula, matrix forms, gener- alizations and properties linked to these numbers were also presented. Finally, this article makes it possible to contribute to the mathematical field and pro- vides mathematical researchers with knowledge about Leonardo’s sequence and its evolutionary process. References [1] F.R.V. Alves and R.P.M. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab, Int. Elect. J. Math. Ed. 15 (2020), no. 2, Article No. em0575, 9 pp. [2] F.R.V. Alves, R.P.M. Vieira, and P.M.M.C. Catarino, Visualizing the Newtons fractal from the recurring linear sequence with Google Colab: An example of Brazil X Portugal research, Int. Elect. J. Math. Ed. 15 (2020), no. 3, Article No. em0594, 19 pp. [3] P. Catarino and A. Borges, On Leonardo numbers, Acta Math. Univ. Comenian. (N.S.) 89 (2020), no. 1, 75–86. [4] C.J. Harman, Complex Fibonacci numbers, Fibonacci Quart. 19 (1981), no. 1, 82–86. [5] A. Karataş and S. Halici, Horadam octonions, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 25 (2017), no. 3, 97–106. [6] O. Keçilioğlu and I. Akkus, The Fibonacci octonions, Adv. Appl. Clifford Algebr. 25 (2015), no. 1, 151–158. [7] A.G. Shannon, A note on generalized Leonardo numbers, Notes Number Theory Dis- crete Math. 25 (2019), no. 3, 97–101. [8] R.P.M. Vieira, F.R.V. Alves, and P.M.M.C. Catarino, Relações bidimensionais e iden- tidades da sequência de Leonardo, Revista Sergipana de Matemática e Educação Matemática 4 (2019), no. 2, 156–173. [9] R.P.M. Vieira, F.R.V. Alves, and P.M.M.C. Catarino, Uma extensão dos octônios de Padovan para inteiros não positivos, C.Q.D. – Revista Eletrônica Paulista de Matemática 19 (2020), Edição Dezembro, 9–16. [10] R.P.M. Vieira, M.C. dos S. Mangueira, F.R.V. Alves, and P.M.M.C. Catarino, A forma matricial dos números de Leonardo, Ci. e Nat. 42 (2020), 40 yrs. – Anniv. Ed., Article No. e100, 6 pp. The generalization of Gaussians and Leonardo’s octonions 137 Renata Passos Machado Vieira Federal University of Ceará Fortaleza-CE Brasil e-mail: re.passosm@gmail.com Milena Carolina dos Santos Mangueira Federal Institute of Ceará Fortaleza-CE Brasil e-mail: milenacarolina24@gmail.com Francisco Régis Vieira Alves Federal Institute of Ceará Fortaleza-CE Brasil e-mail: fregis@ifce.edu.br Paula Maria Machado Cruz Catarino University of Trás-os-Montes and Alto Douro Vila Real Portugal e-mail: pcatarino23@gmail.com

Journal

Annales Mathematicae Silesianae – de Gruyter

Published: Mar 1, 2023

Keywords: matrix form; Binet’s formula; Gaussian; octonions; Leonardo sequence; 11B37; 11B39

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