Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 2022, Volume 28, Number 1, 115–123 DOI: 10.7546/nntdm.2022.28.1.115-123 Leonardo’s bivariate and complex polynomials Milena Carolina dos Santos Mangueira1, Renata Passos Machado Vieira2, Francisco Regis Vieira Alves3 and Paula Maria Machado Cruz Catarino4 1 Department of Mathematics, Federal Institute of Education, Science and Technology of State of Ceara - IFCE Treze of Maio, Brazil e-mail: milenacarolina24@gmail.com 2 Department of Mathematics, Federal Institute of Education, Science and Technology of State of Ceara - IFCE Treze of Maio, Brazil e-mail: re.passosm@gmail.com 3 Department of Mathematics, Federal Institute of Education, Science and Technology of State of Ceara - IFCE Treze of Maio, Brazil e-mail: fregis@gmx.fr 4 University of Trás-os-Montes and Alto Douro - UTAD Vila Real, Portugal e-mail: pcatarin@utad.pt Received: 25 February 2021 Revised: 17 January 2022 Accepted: 27 February 2022 Online First: 28 February 2022 Abstract: Given the purpose of mathematical evolution of LeonardoŠs sequence, we have the prospect of introducing complex polynomials, bivariate polynomials and bivariate polynomials around these numbers. Thus, this paper portrays in detail the insertion of the variable x, y and the imaginary unit i in the sequence of Leonardo. Nevertheless, the mathematical results from this process of complexiĄcation of these numbers are studied, correlating the mathematical evolution of that sequence. Keywords: Leonardo complex bivariate polynomials, Leonardo polynomials, Leonardo sequence. 2020 Mathematics Subject Classification: 11B37, 11B69. 115 1 Introduction LeonardoŠs sequence was initially presented by Catarino and Borges [5]. Historically, it is believed that these numbers have been studied by Leonardo de Pisa, known as Leonardo Fibonacci, and therefore not proven in any work in the literature, due to the scarcity of research [3]. This sequence has been studied and evolved mathematically, as we can see in the works of [2, 7Ű9]. Thus, we have the Leonardo sequence satisfying the following recurrence relationship: Len = Ln−1 + Ln−2 + 1, n ≥ 2. (1) And yet, for n + 1 one can rewrite this recurrence relationship as Ln+1 = Ln + Ln−1 + 1. Also, subtracting Ln − Ln+1 gives another recurrence relation for this sequence. Ln − Ln+1 = Ln−1 + Ln−2 + 1 − Ln − Ln−1 − 1, Ln+1 = 2Ln − Ln−2 , (2) where L0 = L1 = 1 are the initial conditions. Thus, the initial values of the sequence are as follows: 1, 1, 3, 5, 9, 15, 25, . . . . In order to continue the mathematical evolutionary process of LeonardoŠs numbers, in this paper, we will present a study around LeonardoŠs numbers in their polynomial, bivariate polynomial and complex bivariate polynomial form. We can Ąnd sequences in their polynomial form in works presented in the literature of pure mathematics, and yet, according to [6] the complex bivariate polynomials encompasses the polynomial terms of the studied sequence in an evolutionary process of its algebraic form. That is, Ąrst, polynomials are considered with one variable and two variables, then the imaginary component i is inserted, then these polynomials are explored in their complex form. 2 Leonardo’s polynomials Based on the Fibonacci polynomials, studied in 1883 by the mathematicians Eugène Catalan (1814Ű1894) and Ernst Erich Jacobsthal (1881Ű1965) [1], one can then introduce LeonardoŠs polynomials. Definition 2.1. LeonardoŠs polynomials, ln (x), for n ≥ 3 are given by: ln (x) = 2xln−1 (x) − ln−3 (x), with l0 (x) = l1 (x) = 1 and l2 (x) = 3. The Ąrst terms of the sequence are given in the following Table 1. 116 n ln (x) 0 1 2 3 4 5 6 .. . 1 1 3 6x − 1 12x2 − 2x − 1 24x3 − 4x2 − 2x − 3 48x4 − 8x3 − 4x2 − 12x + 1 .. . Table 1. First terms of LeonardoŠs polynomial sequence Theorem 2.2. The matrix form of LeonardoŠs polynomials, for n ≥ 2 and with n ∈ N, is given by: n  2x 1 0 h i h i   3 1 1  0 0 1 = ln+2 (x) ln+1 (x) ln (x) . −1 0 0 Proof. We use the principle of Ąnite induction. For n = 2, we have that: 2  2x 1 0 h i h i h i   3 1 1  0 0 1 = 12x2 − 2x − 1 6x − 1 3 = l4 (x) l3 (x) l2 (x) . −1 0 0 Validating equality. Assuming it is valid for n = k, k  h i 2x  3 1 1 0 −1 ∈ N, we have that: k 1 0 h i  0 1 = lk+2 (x) lk+1 (x) lk (x) . 0 0 Now, verifying that it is valid for n = k + 1, we have that:    k   2x 1 0 h i 2x 1 0 h i 2x 1 0       3 1 1  0 0 1  0 0 1 = lk+2 (x) lk+1 (x) lk (x)  0 0 1 −1 0 0 −1 0 0 −1 0 0 h i = 2xlk+2 (x) − lk (x) lk+2 (x) lk+1 (x) h i = lk+3 (x) lk+2 (x) lk+1 (x) . The characteristic equation of this Leonardo polynomial sequence, is given by t3 − 2xt2 + 1 = 0, where x is the polynomial variable. So, we have to t1 , t2 and t3 are the roots of the characteristic equation. 117 Theorem 2.3. BinetŠs formula of LeonardoŠs polynomials, with n ∈ Z, is given by: ln (x) = αtn1 + βtn2 + γtn3 , where t1 , t2 , t3 are the roots of the characteristic equation t3 − 2xt2 + 1 = 0 and α= 3 + (−t2 − t3 ) + t2 t3 3 + (−t1 − t3 ) + t1 t3 3 + (−t1 − t2 ) + t1 t2 , β= 2 , γ= 2 . 2 t1 − t1 t2 − t1 t3 + t2 t3 t2 − t2 t3 − t1 t2 + t1 t3 t3 + t1 t2 − t1 t3 − t2 t3 Proof. Through the Binet formula ln = αtn1 + βtn2 + γtn3 and the recurrence of LeonardoŠs polynomials ln (x) = 2xln−1 (x)−ln−3 (x), with the initial values l0 (x) = l1 (x) = 1 and l2 (x) = 3, it is possible to obtain the following system of equations:   =1  α+β+γ αt1 + βt2 + γt3 = 1 .   αt2 + βt2 + γt2 = 3 1 2 3 Solving the system, we have that: 3 + (−t2 − t3 ) + t2 t3 , t21 − t1 t2 − t1 t3 + t2 t3 3 + (−t1 − t3 ) + t1 t3 , β= 2 t2 − t2 t3 − t1 t2 + t1 t3 3 + (−t1 − t2 ) + t1 t2 γ= 2 . t3 + t1 t2 − t1 t3 − t2 t3 α= Theorem 2.4. The generating function of LeonardoŠs polynomial sequence, for n ∈ N, is given by: ∞ X 1 − 5t + t2 g(ln (x), t) = ln (x)tn = (1 − 2xt + t3 ) n=0 Proof. Let g(ln (x), t) be the generating function of LeonardoŠs polynomial sequence ln (x), then: g(ln (x), t) − g(ln (x)2xt + g(ln (x)t3 = l0 (x) + (l1 (x) − 2l0 (x))t + (l2 (x) − 2l1 (x))t2 , g(ln (x), t)(1 − 2xt + t3 ) = 1 − 5t + t2 , g(ln (x), t) = 1 − 5t + t2 . (1 − 2xt + t3 ) 3 Leonardo’s bivariate polynomials In this section, LeonardoŠs bivariate polynomials will be introduced. The Ąrst terms of this sequence are given in Table 2. Definition 3.1. LeonardoŠs bivariate polynomials, ln (x, y), for n ≥ 3 are given by: ln (x, y) = 2xln−1 (x, y) − yln−3 (x, y), with l0 (x, y) = l1 (x, y) = 1 and l2 (x, y) = 3. 118 n ln (x, y) 0 1 2 3 4 5 6 .. . 1 1 3 6x − y 12x2 − 2xy − y 24x3 − 4x2 y − 2xy − 3y 48x4 − 8x3 y − 4x2 y − 12xy + y 2 .. . Table 2. First terms of LeonardoŠs bivariate polynomial sequence It is observed that with the values x = y = 1, we have LeonardoŠs original sequence, as shown in Table 3. n ln (1, 1) 0 1 2 3 4 5 6 .. . 1 1 3 5 9 15 25 .. . Table 3. First terms of LeonardoŠs bivariate polynomial sequence Theorem 3.2. The matrix form of LeonardoŠs bivariate polynomials, for n ≥ 2 and with n ∈ N, is given by: n  2x 1 0 h i h i   3 1 1  0 0 1 = ln+2 (x, y) ln+1 (x, y) ln (x, y) . −y 0 0 Proof. Analogously to the proof of Theorem 2.2, the present theorem can be validated. The characteristic equation of LeonardoŠs bivariate polynomial sequence is given by q −2xq 2 +y = 0, on what x and y are the polynomial variables. So, we have q1 , q2 and q3 . 3 Theorem 3.3. BinetŠs formula of LeonardoŠs bivariate polynomials, with n ∈ Z, is given by: ln (x, y) = αq1n + βq2n + γq3n , on what q1 , q2 , q3 are the roots of the characteristic equation q 3 − 2xq 2 + y = 0 and α= 3 + (−q1 − q3 ) + q1 q3 3 + (−q1 − q2 ) + q1 q2 3 + (−q2 − q3 ) + q2 q3 , β= 2 , γ= 2 . 2 q1 − q 1 q 2 − q 1 q 3 + q2 q 3 q2 − q 2 q3 − q 1 q2 + q 1 q3 q 3 + q1 q2 − q1 q3 − q2 q 3 119 Proof. Through the Binet formula ln (x, y) = αq1n + βq2n + γq3n and the recurrence of LeonardoŠs bivariate polynomials ln (x, y) = 2xln−1 (x, y) − yn−3 (x, y), with the initial values l0 (x, y) = l1 (x, y) = 1 and l2 (x, y) = 3, it is possible to obtain the following system of equations:   =1  α+β+γ αq1 + βq2 + γq3 = 1   αq 2 + βq 2 + γq 2 = 3 1 2 3 Solving the system, we have that: 3 + (−q2 − q3 ) + q2 q3 , q12 − q1 q2 − q1 q3 + q2 q3 3 + (−q1 − q3 ) + q1 q3 , β = 2 q2 − q 2 q 3 − q 1 q 2 + q1 q 3 3 + (−q1 − q2 ) + q1 q2 . γ = 2 q3 + q 1 q2 − q 1 q 3 − q2 q 3 α = Theorem 3.4. The generator function of LeonardoŠs bivariate polynomial sequence, for n ∈ N, is given by: g(ln (x, y), t) = ∞ X ln (x, y)tn = n=0 1 − 5t + t2 . (1 − 2xt + yt3 ) Proof. Be g(ln (x, y), t) the generating function of LeonardoŠs polynomial sequence ln (x, y), then: g(ln (x, y), t) − g(ln (x, y)2xt + g(ln (x, y)yt3 = l0 (x, y) + (l1 (x, y) − 2l0 (x, y))t + (l2 (x, y) − 2l1 (x, y))t2 g(ln (x, y), t)(1 − 2xt + yt3 ) = 1 − 5t + t2 g(ln (x, y), t) = 1 − 5t + t2 . (1 − 2xt + yt3 ) 4 Leonardo’s complex bivariate polynomials In this section, LeonardoŠs complex bivariate polynomials will be introduced. Definition 4.1. LeonardoŠs complex bivariate polynomials, ln (ix, y), for n ≥ 3 are given by: ln (ix, y) = 2xiln−1 (ix, y) − yln−3 (ix, y), with l0 (ix, y) = l1 (ix, y) = 1, l2 (ix, y) = 3 and i2 = −1. The Ąrst terms of this sequence are given in the following Table 4. 120 n ln (ix, y) 0 1 2 3 4 5 6 .. . 1 1 3 6xi − y −12x2 − 2xyi − y −24x3 i − 2xyi + 4x2 y − 3y 48x4 + 4x2 y + y 2 − 12xyi + 8x3 yi .. . Table 4. First terms of LeonardoŠs complex bivariate polynomial sequence Theorem 4.2. The matrix form of LeonardoŠs complex bivariate polynomials, for n ≥ 2 and with n ∈ N, is given by: n  h i h i 2xi 1 0   3 1 1  0 0 1 = ln+2 (ix, y) ln+1 (ix, y) ln (ix, y) . −y 0 0 Proof. Analogously to the proof of Theorem 2.2, the present theorem can be validated. The characteristic equation of LeonardoŠs complex bivariate polynomial sequence is given by v − 2xiv 2 + y = 0, on what x and y are the polynomial variables. So, we have to v1 , v2 and v3 are the roots of the characteristic equation. 3 Theorem 4.3. BinetŠs formula of LeonardoŠs complex bivariate polynomials, with n ∈ Z, is given by: ln (ix, y) = αv1n + βv2n + γv3n , on what v1 , v2 , q3 are the roots of the characteristic equation v 3 − 2xiv 2 + v = 0 and α= 3 + (−v2 − v3 ) + v2 v3 3 + (−v1 − v3 ) + v1 v3 3 + (−v1 − v2 ) + v1 v2 , β= 2 , γ= 2 . 2 v1 − v1 v2 − v1 v3 + v2 v3 v2 − v2 v3 − v1 v2 + v1 v3 v3 + v1 v2 − v1 v3 − v2 v3 Proof. Through the Binet formula ln (ix, y) = αv1n + βv2n + γv3n and the recurrence of LeonardoŠs complex bivariate polynomials ln (ix, y) = 2xiln−1 (ix, y) − yn−3 (ix, y), with the initial values l0 (ix, y) = l1 (ix, y) = 1 and l2 (ix, y) = 3, it is possible to obtain the following system of equations:   =1  α+β+γ αv1 + βv2 + γv3 = 1 .   αv 2 + βv 2 + γv 2 = 3 1 2 3 Solving the system, we have that: 3 + (−v2 − v3 ) + v2 v3 , v12 − v1 v2 − v1 v3 + v2 v3 3 + (−v1 − v3 ) + v1 v3 , β = 2 v2 − v2 v3 − v1 v2 + v1 v3 3 + (−v1 − v2 ) + v1 v2 γ = 2 . v3 + v1 v2 − v1 v3 − v2 v3 121 α = Theorem 4.4. The generating function of LeonardoŠs complex bivariate polynomial sequence, for n ∈ N, is given by: g(ln (ix, y), t) = ∞ X ln (ix, y)tn = n=0 1 − 5t + t2 . (1 − 2ixt + yt3 ) Proof. Let g(ln (ix, y), t) be the generating function of LeonardoŠs complex bivariate polynomial sequence ln (ix, y), then: g(ln (ix, y), t) − g(ln (ix, y)2ixt + g(ln (ix, y)yt3 = l0 (ix, y) + (l1 (ix, y) − 2l0 (ix, y))t + (l2 (ix, y) − 2l1 (ix, y))t2 , g(ln (ix, y), t)(1 − 2ixt + yt3 ) = 1 − 5t + t2 , g(ln (ix, y), t) = 1 − 5t + t2 . (1 − 2ixt + yt3 ) 5 Conclusion This work presents a study around the Leonardo sequence, continuing the mathematical evolutionary process of this sequence, we present its polynomial form, its bivariate polynomial form and its complex bivariate polynomial form. LeonardoŠs sequence numbers were worked on functions of variables and explored in its complex form after the insertion of the imaginary component i. 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