Appl. Math. Inf. Sci. 14, No. 4, 1-14 (2020) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/AMIS submission quintana ramirez urieles FINAL VERSION Euler Matrices and their Algebraic Properties Revisited Yamilet Quintana1,∗, William Ram´ırez 2 and Alejandro Urieles3 1 Department of Pure and Applied Mathematics, Postal Code: 89000, Caracas 1080 A, Simon Bol´ıvar University, Venezuela 2 Department of Natural and Exact Sciences, University of the Coast - CUC, Barranquilla, Colombia 3 Faculty of Basic Sciences - Mathematics Program, University of Atl´antico, Km 7, Port Colombia, Barranquilla, Colombia Received: 2 Sep. 2019, Revised: 23 Feb. 2020, Accepted: 13 Apr. 2020 Published online: 1 Jul. 2020 Abstract: This paper addresses the generalized Euler polynomial matrix E (α)(x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α)(x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices. Keywords: Euler polynomials, Euler matrix, generalized Euler matrix, generalized Pascal matrix, Fibonacci matrix, Lucas matrix. 1 Introduction Thus, the numbers En := En(0) are also known in pieces of literature as Euler numbers (cf., e.g., [15,21]). The first The classical Euler polynomials En(x) and the generalized six generalized Euler polynomials are Euler polynomials E(α)(x) of (real or complex) order α, n α are usually defined, as follows (see, for details, [1,16,21, (α) (α) E0 (x)=1, E1 (x)= x − , 23]): 2 (α) 2 α(α − 1) E2 (x)=x − αx + , α ∞ n 4 2 xz (α) z α 2 e = En (x) , |z| < π, 1 := 1, 3α 3α(α − 1) α (α − 3) ez + 1 ∑ n! E(α)(x)=x3 − x2 + x − , n=0 3 2 4 8 (1) 2 and (α) 4 3 3α(α − 1) 2 α (α − 3) (1) E4 (x)=x − 2αx + x − x En(x) := En (x), n ∈ N0, (2) 2 2 2 N N N α(α − 1)(α − 5α − 2) where 0 := ∪{0} and = {1,2,3,...}. + , (α) (α) 16 The numbers En := En (0) are called generalized 5α 5α(α − 1) 5α2(α − 3) Euler numbers of order α. The classical Euler numbers εn E(α)(x)=x5 − x4 + x3 − x2 are defined by the generating function 5 2 2 4 5α(α − 1)(α2 − 5α − 2) 2 ∞ zn + x = εn . (3) 16 ez + e−z ∑ n! n=0 α2(α3 − 10α2 + 15α + 10) − . From (1)-(3), it is easy to check that the connection 32 between the classical Euler numbers and the Euler polynomials is given by the formula For a broad information on old literature and new research trends on these classes of polynomials, we 1 strongly encourage the interested reader to [5,8,9,10,15, ε = 2nE , n ∈ N . (4) n n 2 0 16,17,18,19,20,21,22,24,25,26,27]. ∗ Corresponding author e-mail: [email protected] c 2020 NSP Natural Sciences Publishing Cor. 2 Y. Quintana et al.: Euler matrices and their algebraic properties revisited From the generating relation (1), it is fairly 2 Background and previous results straightforward to deduce the addition formula: Throughout this paper, all matrices are in M R , the n n+1( ) (α+β ) n (α) (β ) set of all (n + 1)-square matrices over the real field. Also, En (x + y)= E (x)E (y). (5) ∑ k k n−k for i, j any nonnegative integers, we adopt the following k=0 convention And, it follows that i (α) (α) (α−1) = 0, whenever j > i. En (x + 1)+ En (x)= 2En (x). (6) j (0) n Since En (x)= x , making the substitution β = 0 into (5) In this section, we recall the definitions of the and interchanging x and y, we get generalized Pascal matrix, Fibonacci matrix and, Lucas n matrix. (α) n (α) n−k En (x + y)= E (y)x . (7) ∑ k k Definition 1.Let x be any nonzero real number. The k=0 generalized Pascal matrix of first type P[x] ∈ Mn+1(R) is As an immediate consequence, we have the matrix whose entries are given by (see [2,29]): n n n−k i En(x + y)= Ek(y)x , (8) i− j ∑ j x , i ≥ j, k=0 k pi, j(x)= n n n−k 0, otherwise. En(x)= ∑ Ek x . (9) k=0 k In [2,29,31], some properties of the generalized Using (4), (8) and the well known relation n Pascal matrix of first type are shown, for example, its En(1 − x) = (−1) En(x), it is possible to deduce the matrix factorization by special summation matrices, its following connection formula between En and the associated differential equation and its bivariate classical Euler numbers εn: extensions. The following proposition summarizes some 1 n n algebraic and differential properties of P[x]. − 2n ∑k=0 k εn−k, if n is odd, E = (10) n Proposition 1.Let P[x] ∈ M (R) be the generalized 0, if n is even . n+1 Pascal matrix of first type. Then, the following statements hold. Inspired by the article [32] in which the authors introduce the generalized Bernoulli matrix and show (a)Special value. If the convention 00 = 1 is adopted, then some of its algebraic properties, we examine some it is possible to define properties of generalized Euler matrix. The outline of the paper is, as follows: Section 2 has P[0] := In+1 = diag(1,1,...,1), (11) an auxiliary character and provides some background as well as some results which will be used throughout the where In+1 denotes the identity matrix of order n + 1. paper. In Section 3, we introduce the generalized Euler (b)P[x] is an invertible matrix and its inverse is given by matrix and investigate some interesting particular cases of −1 −1 this matrix, namely the Euler polynomial matrix, the P [x] := (P[x]) = P[−x]. (12) Euler matrix, and the specialized Euler matrix. The main (c)[2, Theorem 2] Addition theorem of the argument. For results of this section are Theorems 1, 2, 3 and 4, which x,y ∈ R, we have contain the information concerning the product formula for the Euler matrix, an explicit expression for the inverse P[x + y]= P[x]P[y]. matrix of the specialized Euler matrix, the factorization of the Euler matrix via the generalized Pascal matrix of first (d)[2, Theorem 5] Differential relation (Appell type kind, and a useful factorization for the inverse matrix of a polynomial entries). P[x] satisfies the following particular “horizontal sliding” of the Euler polynomial differential equation matrix, respectively. Section 4 shows some factorizations of the generalized Euler matrix in terms the Fibonacci and DxP[x]= LP[x]= P[x]L, Lucas matrices, respectively (cf. Theorems 5 and 6). where D P x is the matrix resulting from taking the Also, some new identities involving Fibonacci and Lucas x [ ] derivative with respect to x of each entry of P x and numbers are presented in this section. In Section 5, we [ ] the entries of the n 1 n 1 matrix L are given provide some factorizations of the Euler polynomial ( + ) × ( + ) by matrix in terms of Stirling matrices, and the shifted Euler ′ matrices as well as their connection with Vandermonde pi, j(0), i ≥ j, j + 1, i = j + 1, matrices. Section 6 is devoted to conclusion and further li, j = = research. , . 0, otherwise, 0 otherwise c 2020 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 14, No. 4, 1-14 (2020) / www.naturalspublishing.com/Journals.asp 3 (e)([29, Theorem 1]) The matrix P[x] can be factorized, For x any nonzero real number, the following relation as follows: between the matrices P[x] and L was stated and proved in [33, Theorem 3.1]. P[x]= Gn[x]Gn−1[x]···G1[x], (13) LG H L where Gk[x] is the (n + 1) × (n + 1) summation matrix P[x]= [x]= [x] , (14) given by where the entries of the (n+1)×(n+1) matrices G [x] and I 0 H n−k , k = 1,...,n − 1, [x] are given by 0 Sk[x] Gk[x]= − j−1 i+1 i i i − 1 g , (x)=x x − 3x Sn[x], k = n, i j j j being S [x] the (k + 1) × (k + 1) matrix whose entries − j−1 i+1 i−1 x k + x 5(−1) 2 mi−1, j+1 , Sk(x;i, j) are given by 2 h i xi− j, j ≤ i, S x;i, j 0 i, j k . − j−1 i+1 i i i k( )= ( ≤ ≤ ) hi, j(x)=x x − 3x , > , j j + 1 0 j i 5xi+ j+2 2 Another necessary structured matrices in what follows, + x− j−1 (−1) j+1 n , 2 j+3 i+1, j+3 x are the Fibonacci and Lucas matrices. Below, we recall the definitions of each matrix. respectively, with Definition 2.Let {Fn}n≥1 be the Fibonacci sequence, i.e., i k k k Fn = Fn−1 + Fn−2 for n ≥ 2 with initial conditions F0 = 0 ∑ (−1) x , i ≥ j, F R k= j j and F1 = 1.