Euler Matrices and Their Algebraic Properties Revisited

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. The Fibonacci matrix ∈ Mn+1( ) is the mi, j(x) := and matrix whose entries are given by [13]:   0, i < j, Fi− j+1, i − j + 1 ≥ 0,  i k i k f = ∑ (−1) x , i ≥ j, i, j  k= j k 0, i − j + 1 < 0. ni, j(x) :=   0, i < j. −1 ˜  Let F be the inverse of F and denote by fi, j the entries of F −1.In[13], the authors obtained the following  explicit expression for F −1. 3 The generalized Euler matrix 1, i = j, Definition 4.The generalized (n + 1) × (n + 1) Euler ˜ α fi, j = −1, i = j + 1, j + 2, matrix E ( )(x) is defined by   , . i (α) 0 otherwise E (x), i ≥ j,  (α) j i− j  E , (x)= Definition 3.Let {Ln}n≥1 be the Lucas sequence, i.e. i j  Ln+2 = Ln+1 + Ln for n ≥ 1 with initial conditions L1 = 1  0, otherwise. and L2 = 3. The Lucas matrix L ∈ Mn+1(R) is the matrix whose entries are given by [33]: While, E (x) := E (1)(x)and E := E (0) are called the Euler polynomial matrix and the Euler matrix, respectively. In Li− j+1, i − j ≥ 0, 1 E E 1 the particular case x = 2 , we call the matrix := 2 li, j = specialized Euler matrix.  0, otherwise.  −1 ˜ It is clear that (6) yields the following matrix identity: Let L be the inverse of L and denote by li, j the L −1 entries of .In[33, Theorem 2.2], the authors obtained E (α)(x + 1)+ E (α)(x)= 2E (α−1)(x). (15) the following explicit expression for L −1. 1, i = j, Since E (0)(x)= P[x], replacing α by1in(15), we have

 −3, i = j + 1, E (x + 1)+ E (x)= 2P[x]. (16)  l˜i, j =   5(−1)i− j2i− j−2, i ≥ j + 2, Then, putting x = 0 in (16) and taking into account  (11), we get E E  0, otherwise. (1)+ = 2In+1.    c 2020 NSP Natural Sciences Publishing Cor. 4 Y. Quintana et al.: Euler matrices and their algebraic properties revisited

Analogously, Proof.The application of induction on k gives the desired result. E + E (−1)= 2P[−1].

From (4), it follows that the entries of the specialized Taking x = x1 = x2 = ··· = xk and α = α1 = α2 = ··· = Euler matrix E are given by αk, we obtain the following simple formula for the powers of the generalized Euler matrix, and consequently, for the i j−i j 2 εi− j, i ≥ j, powers of the Euler polynomial and Euler matrices. ei, j = (17)  Corollary 2.The generalized Euler matrix E (α)(x) 0, otherwise.  satisﬁes the following identity. From (10), it follows that the entries of the Euler k  (α) (kα) matrix E are given by E (x) = E (kx).

i In particular, j Ei− j, i > j and i − j odd, (E (x))k =E (k)(kx),  (19) Ei, j =  1, i = j, E k =E (k).   0, otherwise. Remark.Note that Theorem 1 and Corollaries 1 and 2 are  respectively,the analogous of Theorem 2.1 and Corollaries  The next result is an immediate consequence of 2.2 and 2.3 of [32] in the setting of Euler matrices. Definition 4 and the addition formula (5). Let D ∈ Mn+1(R) be the matrix whose entries are defined by Theorem 1.The generalized Euler matrix E (α)(x) satisfies the following product formula. i− j i j−i−1 (1 + (−1) ) j 2 , i ≥ j, (α+β ) (α) (β ) (β ) (α) di, j = E (x + y)= E (x)E (y)= E (x)E (y)   0, otherwise. = E (α)(y)E (β )(x). (18) Theorem 2.The inverse matrix of the specialized Euler Proof.We proceed as in the proof of [32, Theorem 2.1], matrix E is given by (α,β ) , making the corresponding modifications. Let Ai, j (x y) E−1 = D. be the (i, j)-th entry of the matrix product E (α)(x)E (β )(y). Then, by the addition formula (5), we have Furthermore, n −1 , i k (k) k k A(α β )(x,y)= E(α)(x) E(β ) (y) E = D . i, j ∑ k i−k j k− j 2 k=0 i i k Proof.Taking into account (4) and (17), it is possible to = E(α)(x) E(β ) (y) ∑ k i−k j k− j deduce k= j n k (1 + (−1) ) n n−k 1 i 2 En−k i i − j (α) (β ) ∑ 2 k 2 = E (x)E (y) k=0 ∑ j i − k i−k k− j k= j n (1 + (−1)k) n = εn−k = δn,0, i i− j i − j ∑ 2 k = E(α) (x)E(β )(y) k=0 j ∑ k i− j−k k k=0 where δn,0 is the Kronecker delta (cf., e.g., [19, pp. i α β 107-109]). Hence, we obtain that the (i, j)-th entry of the = E( + )(x + y), DE j i− j matrix product may be written as i i−k which implies the first equality of (18). The second and i (1 + (−1) ) k−i k 1 ∑ 2 Ek− j third equalities of (18) can be derived in a similar way. k= j k 2 j 2 i i−k i j−i i − j (1 + (−1) ) k− j 1 = 2 2 Ek− j Corollary 1.Let (x ,...,x ) ∈ Rk. For α real or complex j ∑ k − j 2 2 1 k j k= j E (α j) parameters, the Euler matrix (x) satisfies the i− j i− j−k following product formula, j = 1,...,k. i j−i i − j (1 + (−1) ) k 1 = 2 ∑ 2 Ek j k=0 k 2 2 k E (α1+α2+···+αk)(x + x + ··· + x )= E (α j)(x ). i j−i 1 2 k ∏ j = 2 δi− j,0, j=1 j

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and consequently, DE = In+1. Similar arguments allow to Remark.Note that the relation (21) is the analogous of −1 show that ED = In+1, so E = D. [32, Eq. (13)] in the context of Euler polynomial matrices Finally, from the identity E−1 = D and (19) we see and, the counterpart of (22) is [32, Eq. (14)]. However, that the relation (23) is slightly different from [32, Eq. (14)], since it involves an Euler polynomial matrix with “shifted −1 k −1 k argument” and the specialized Euler matrix. More E (k) = Ek = E−1 = D k. 2 precisely, the relation [32, Eq. (14)] reads as This last chain of equalities finishes the proof. B(x)= P[x]B. Consequently, this relation expresses the Bernoulli The calculation of E−1 depends on the use of inverse polynomial matrix B(x) in terms of the matrix product relations derived from exponential generating functions between the generalized Pascal matrix of first type P[x] (cf. [19, Chap. 3, Sec. 3.4]). This tool can be applied to and the Bernoulli matrix B. While, on the left hand side define E−1, but it does not work for determining of E −1. of (23), an Euler polynomial matrix with “shifted This fact and (10) suggest that methodology proposed in argument” appears, and the matrix product on the right [32] does not suffice to find an explicit formula for E −1. hand side of (23) contains the specialized Euler matrix E. The next result establishes the relation between the generalized Euler matrix and the generalized Pascal The following example shows validity of Theorem 3. matrix of first type. Example 1.Let us consider n = 3. It follows from the Theorem 3.The generalized Euler matrix E (α)(x) satisfies definition 4 that the following relation. 1 000 1 0 00 α α α E ( )(x + y)= E ( )(x)P[y]= P[x]E ( )(y) 0 100 1 x 1 00 E E (α) =  1 , x + =  2 1 . = E (y)P[x]. (20) − 4 0 10 2 x − 4 2x 1 0 3 3 3  0 − 01 x3 − x 3x2 − 3x 1 In particular,  4   4 4      On the other hand, from (23) and a simple E E E (x + y)= P[x] (y)= P[y] (x), (21) computation, we have

1 0 00 1 000 E (x)= P[x]E , (22) 0 100 E 1 x 1 00 x + =  2  1  2 x 2x 1 0 − 4 0 10 3 2 3 1 x 3x 3x 1  0 − 01 E x + = P[x]E, (23)   4  2    P[x] E and | 1{z 0} 00 1 | {z } E = P − E. x 1 00 2 =  2 1 . x − 4 2x 1 0 x3 − 3 x 3x2 − 3 3x 1 Proof.The substitution β = 0 into (18) yields  4 4    The next theorem is followed by a simple computation. E (α)(x + y)= E (α)(x)E (0)(y)= E (0)(x)E (α)(y) (α) (0) = E (y)E (x). Theorem 4.The inverse of the Euler polynomial matrix E x + 1 can be expressed as Since E (0)(x)= P[x], we obtain 2 −1 (α) (α) 1 E (x + y)= P[x]E (y). E x + = E−1P[−x]= DP[−x]. (24) 2 A similar argument allows to show that E (α)(x + y)= E (α)(x)P[y] and E (α)(x + y)= E (α)(y)P[x]. In particular, 1 Next, the substitution α = 1 into (20) yields (21). From E −1 = DP . (25) the substitutions y = 0 and y = 1 into (21), we obtain the 2 2 relations (22) and(23), respectively. Finally, the substitution x = − 1 into (23) completes the Proof.Using (12), (23) and Theorem 2, the relation (24) is 2 1 proof. deduced. The substitution x = − 2 into (24) yields (25).

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Example 2.Let us consider n = 3. From the deﬁnition 4 and Corollary 4.For x any nonzero real number, the Euler E 1 a standard computation, we obtain polynomial matrix x + 2 and its inverse can be L −1 factorized, respectively, in terms of the Lucas matrix 1 0 00 and its inverse, as follows: −1 x 1 00 E 1 x + =  2 1  1 2 x − 4 2x 1 0 E LG E H L E 3 3 x + = [x] = [x] , x3 − x 3x2 − 3x 1 2  4 4   1 0 00 −x 1 00 1 −1 =  2 1 . E x D G x −1L −1 DL −1 H x −1. x + 4 −2x 1 0 + = ( [ ]) = ( [ ]) 3 3 2 3 2 −x − 4 x 3x + 4 −3x 1   In particular, On the other hand, from (24), we have  1 1 1000 1 0 00 E =LG − E = H − L E, 1 −1 0100 −x 1 00 2 2 E x + =  1  2  2 4 010 x −2x 1 0 3 3 2 0 01 −x 3x −3x 1 1 1  4   1 − 1 −    E −1 =D G − L −1 = DL −1 H − . D P[−x] 2 2 1 0 00 | {z }| {z } We end this section showing other identities, which can −x 1 00 =  2 1 . be easily deduced from the content of this paper. Thus, we x + 4 −2x 1 0 3 3 2 3 will omit the details of their proofs. −x − 4 x 3x + 4 −3x 1   E L E  1  Dx (x + y)= P[x] (y), Hence, when x = − 2 , we get

1000 1000 1000 DxE (x)=LP[x]E , 1 1 −1 0100 2 100 2 100 E =  1  1  =  1 . 4 010 4 110 2 110 3 1 3 3 1 3 3 1 0 01 1  1 DxE x + =LP[x]E,  4  8 4 2   2 2 2  2  D  1    P[ 2 ] −1 At this point,| we{z should}| refer{z to} the recent work [11] 1 D E x + =DLP[−x]. since it states an explicit formula to the inverse matrix of x 2 the q-Pascal matrix plus one in terms of the q-analogue of the Euler matrix E . As a consequence of the relations (13), (14), and 4 Generalized Euler polynomial matrices via Theorems 3 and 4, we obtain the following corollaries. Fibonacci and Lucas matrices E 1 Corollary 3.The Euler polynomial matrix x + 2 and For 0 ≤ i, j ≤ n and α a real or complex number, let its inverse can be factorized by summation matrices, as M (α)(x) be the (n + 1) × (n + 1) matrix whose entries are follows: given by (cf. [32, Eq. (18)]): 1 E x + =Gn[x]Gn−1[x]···G1[x]E, 2 (α) i (α) i − 1 (α) m˜ (x)= E (x) − E (x) i, j j i− j j i− j−1 1 i 2 1 − − (α) − Ei− j−2(x). (26) E x + =DGn[−x]Gn−1[−x]···G1[−x]. j 2 We denote M x M (1) x and M M 0 . In particular, ( )= ( ) = ( ) Similarly, let N (α)(x) be the (n + 1) × (n + 1) matrix 1 1 1 whose entries are given by (cf. [32, Eq. (32)]): E =Gn − Gn−1 − ···G1 − E, 2 2 2 i i n˜(α)(x)= E(α)(x) − E(α) (x) i, j j i− j j + 1 i− j−1 −1 1 1 1 i (α) E =DGn G ···G . − E (x). 2 n−1 2 1 2 j + 2 i− j−2

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We denote N (x)= N (1)(x) and N = N (0). for i ≥ 2, and From the deﬁnitions of M (α)(x) and N (α)(x), we see ˆ(α,1) (α) (α) that li,1 (x)= iEi−1(x) − 3(i − 1)Ei−2(x) i 2 (α) (α) (α) (α) (α) 5 − m˜ (x)= m˜ (x)= n˜ (x)= n˜ (x)= E (x)= 1, i−k (α) 0,0 1,1 0,0 1,1 0 + ∑ (−2) kEk−1(x), 4 k=1 m˜ (α)(x)= n˜(α)(x)= 0, j ≥ 1, 0, j 0, j for i ≥ 3. The following results show some factorizations of (α) (α) (α) (α) α (α) m˜ 1,0 (x)= n˜1,0 (x)= E1 (x) − E0 (x)= x − 2 − 1, E (x) in terms of Fibonacci and Lucas matrices, respectively. (α) (α) m˜ 1, j (x)= n˜1, j (x)= 0, j ≥ 2, Theorem 5.The generalized Euler polynomial matrix E (α) m˜ (α)(x)= E(α)(x) − E(α)(x) − E(α)(x), i ≥ 2, (x) can be factorized in terms of the Fibonacci i,0 i i−1 i−2 matrix F , as follows: (α) (α) (α) i(i−1) (α) (α) (α) n˜i,0 (x)= Ei (x) − iEi−1(x) − 2 Ei−2(x), i ≥ 2. E (x)= FM (x), (27)

For 0 ≤ i, j ≤ n and α a real or complex number, let or, L (α) E (α) N (α) F 1 (x) be the (n + 1) × (n + 1) matrix whose entries are (x)= (x) . (28) given by In particular, (α,1) i (α) i − j (α) lˆ (x)= E (x) − 3 E (x) FM E N F i, j j i− j j i− j−1 (x)= (x)= (x) , (29) i−2 k = 5 (−1)i−k2i−k−2 E(α) (x). ∑ j k− j FM = E = N F , (30) k= j and L L (1) L L We denote 1(x)= 1 (x) and 1 = 1(0). 1 1 FM E N F L (α) = = . (31) Similarly, let 2 (x) be the (n + 1) × (n + 1) matrix 2 2 whose entries are given by Proof.Since the relation (27) is equivalent to , i i lˆ(α 2)(x)= E(α)(x) − 3 E(α) (x) F −1E (α)(x)= M (α)(x), it is possible to follow the proof i, j j i− j j + 1 i− j−1 given in [32, Theorem 4.1], making the corresponding i i modiﬁcations, for obtaining (27). The relation (28) can be +5 (−1)k− j2k− j−2 E(α)(x). ∑ k i−k obtained using a similar procedure. The relations (29), k= j+1 (30) and (31) are straightforward consequences of (27) L L (1) L L and (28). We denote 2(x)= 2 (x) and 2 = 2(0). L (α) L (α) From the deﬁnitions of 1 (x) and 2 (x), we see that Also, the relations (27)and(28) allow us to deducethe following identity: ˆ(α,1) ˆ(α,2) li,i (x)= li,i (x)= 1, i ≥ 0, M (α)(x)= F −1N (α)(x)F . ˆ(α,1) ˆ(α,2) l0, j (x)= l0, j (x)= 0, j ≥ 1, As a consequence of Theorems 4 and 5, we can derive simple factorizations for the inverses of the polynomial ˆ(α,1) (α) (α) α matrices M x 1 and N x 1 : l1,0 (x)= E1 (x) − 3E0 (x)= x − 2 − 3, + 2 + 2 lˆ(α,2) x E(α) x 11 E(α) x x α 11 , Corollary 5.The inverses of the polynomial matrices 1,0 ( )= 1 ( ) − 2 0 ( )= − 2 − 2 M 1 N 1 x + 2 and x + 2 can be factorized, as follows: (α,1) (α,2) lˆ x lˆ x 0, j 2, −1 1, j ( )= 1, j ( )= ≥ 1 M x + = DP[−x]F , 2 ˆ(α,1) (α) (α) 5 i−2 i−k (α) li,0 (x)= Ei (x) − 3Ei−1(x)+ 4 ∑k=0(−2) Ek (x), 1 −1 (α,2) (α) (α) 5 i i (α) N x + = FDP[−x]. lˆ (x)= E (x) − 3iE (x)+ ∑ (−2)k E (x), 2 i,0 i i−1 4 k=1 k i−k

c 2020 NSP Natural Sciences Publishing Cor. 8 Y. Quintana et al.: Euler matrices and their algebraic properties revisited

In particular, Another identities involving Fibonacci and Lucas numbers as well as the generalized Euler polynomials and 1 1 numbers are, as follows: M −1 = DP F , and N −1 = FDP , 2 2 Theorem 7.For 0 ≤ r ≤ n and α any real or complex 1 −1 1 −1 number, we have M = DF , and N = FD. 2 2 n (r + 1)α + 2 E(α) (x)= F + (r + 1)x − F r n−r n−r+1 2 n−r An analogous reasoning as used in the proof of n Theorem 5 allows us to prove the results below. k (α) k − r (α) + ∑ Ek−r(x) − Ek−r−1(x) k=r+2 r k Theorem 6.The generalized Euler polynomial matrix h k − r − 1 E (α)(x) can be factorized in terms of the Lucas matrix + E(α) (x) F (34) k − 1 k−r−2 n−k+1 L , as follows: α E (α) LL (α) = Fn−r+1 + n x − − 1 Fn−r (x)= 1 (x), (32) 2 n−2 n h ni− k or, + E(α) (x) − E(α) (x) ∑ k n−k k n−k−1 E (α) L (α) L k=0 + 1 (x)= 2 (x) . (33) h n − k − 1 In particular, + E(α) (x) F . k + 2 n−k−2 k−r+1 LL (x)= E (x)= L (x)L , 1 2 Proof.We proceed as in the proof of [32, Theorem 4.2], LL = E = L L , making the corresponding modiﬁcations. From (26), it is 1 2 clear that and 1 1 ( ) ( ) α α (r + 1)α + 2 LL 2 (x)= E = L 2 (x)L . m˜ ( )(x)= 1, m˜ ( ) (x) = (r + 1)x − , 1 2 r,r r+1,r 2 Also, the relations (32)and(33) allow us to deducethe and, for k ≥ r + 2: following identity: k k − r m˜ (α)(x)= E(α)(x) − E(α) (x) L (α)(x)= L −1L (α)(x)L . k,r r k−r k k−r−1 1 2 h k − r − 1 + E(α) (x) . Corollary 6.The inverses of the polynomial matrices k 1 k−r−2 L 1 L 1 − 1 x + 2 and 2 x + 2 can be factorized, as follows: Next, it follows from (27) that −1 1 n L D L n α α α 1 x + = P[−x] , E( ) (x)= E( )(x)= F m˜ ( )(x) 2 r n−r n,r ∑ n−k+1 k,r k=r −1 (α) 1 = Fn−r+1 + Fn−rm˜ (x) L x + = L DP[−x]. r+1,r 2 2 n (α) + ∑ Fn−k+1m˜ k,r (x), In particular, k=r+2 1 1 so, L −1 = DP L , and L −1 = L DP , 1 2 2 2 1 −1 1 −1 L = DL , and L = L D. 1 2 2 2 Remark.If we consider a ∈ C, b ∈ C \{0} and s = 0,1, then Theorems 5 and 6, as well as their corollaries have corresponding analogous forms for generalized Fibonacci matrices of type s, F (a,b,s), and generalized Fibonacci matrices U (a,b,0) with second order recurrent sequence (a,b) Un subordinated to certain constraints [28].

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Theorem 8.For any real or complex number α, we have the following identities n (r + 1)α + 2 E(α) (x)= F + (r + 1)x − F r n−r n−r+1 2 n−r (α) α En (x)= Ln+1 + x − − 3 Ln n k k − r 2 + E(α)(x) − E(α) (x) n ∑ r k−r k k−r−1 + E(α)(x) − 3E(α) (x) L k=r+2 ∑ k k−1 n−k+1 h k=2 k − r − 1 (α) + E (x) Fn k 1. n k−2 k − 1 k−r−2 − + k−s k−s−2 (α) +5 ∑ ∑ (−1) 2 Ln−k+1Es (x), (35) This chain of equalities completes the ﬁrst part of the k=2 s=0 proof. The second one is obtained in a similar way, taking whenever n ≥ 2. into account the following identities: (α) nE (x)= Ln + (2x − α − 3)Ln−1 α n−1 n˜(α)(x)= 1, n˜(α) (x)= n x − − 1, n n,n n,n−1 2 (α) (α) + ∑ kEk−1(x) − 3(k − 1)Ek−2(x) Ln−k+1 k 3 and, for 0 ≤ k ≤ n − 2: = n k−2 k−s k−s−2 (α) (α) n (α) n − k (α) +5 (−1) 2 sL E (x), n˜ (x)= E (x) − E (x) ∑ ∑ n−k+1 s−1 n,k k n−k k + 1 n−k−1 k=3 s=1 h n − k − 1 (36) + E(α) (x) . k + 2 n−k−2 whenever n ≥ 3. Corollary 8.The following identities hold. Corollary 7.For 0 ≤ r ≤ n and α any real number,we have α (−1)nE(α)(x)= L − x − + 3 L n n+1 2 n n n (α) r n (−1) En−r(x) = (−1) Fn−r+1 k (α) (α) r + ∑ (−1) Ek (x)+ 3Ek−1(x) Ln−k+1 k=2 (r + 1)(2x − α)+ 2 + (−1)r+1 F n k−2 2 n−r k−s k−s−2 (α) +5 ∑ ∑ (−1) 2 Ln−k+1Es (x), n k k − r k=2 s=0 + (−1)k E(α)(x)+ E(α) (x) (37) ∑ r k−r k k−r−1 k=r+2 ( " whenever n ≥ 2. k − r − 1 (α) E x F n (α) − k−r−2( ) n−k+1 (−1) −1nE (x)= L + (α − 2x − 3)L k − 1 #) n−1 n n−1 n k−1 (α) + ∑ (−1) kEk−1(x) r r+1 α k=3 = (−1) Fn−r+1 + (−1) n x − − 1 Fn−r 2 (α) +3(k − 1)Ek 2(x) Ln−k+1 n−2 h i − n (α) n − k (α) + (−1)n−k+r E (x)+ E (x) n k−2 ∑ k n−k k + 1 n−k−1 k−1 k−s−2 (α) k=0 ( " +5 ∑ ∑ (−1) 2 sLn−k+1Es−1(x), k=3 s=1 n − k − 1 (α) (38) + En−k−2(x) Fk−r+1. k + 2 #) whenever n ≥ 3. Proof.Replacing x by α − x in (34) and applying the By (35),(36),(37) and (38), we obtain the following formula (α) n (α) interesting identities involving Lucas and Euler numbers. En (x) = (−1) En (α − x) to the resulting identity, we obtain the ﬁrst identity of –For n ≥ 2: Corollary 7. An analogous reasoning yields the second 7 n identity. E − L − L = E − 3E n n+1 2 n ∑ k k−1 k=2 k−2 Analogous reasonings to those used in the proofs of k−s k−s−2 Theorem 7 and Corollary 7 allow us to prove the following =+5 ∑ (−1) 2 Es Ln−k+1, s=0 ! results.

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A recent connection between the Stirling numbers of 5 the second type and the Euler polynomials is given by the (−1)nE = L − L n n+1 2 n formula (see [7, Theorem 3.1, Eq. (3.3)]): n n n−k+1 l 1 k n (−1) − (l − 1)! + ∑ (−1) (Ek + 3Ek−1)Ln−k+1 E (x)= (−1)n−k n ∑ k ∑ 2l−1 k=2 k=0 " l=1 n k−2 k−s k−s−2 +5 ∑ ∑ (−1) 2 Ln−k+1Es. ×S(n − k + 1,l) xk. (40) k=2 s=0 # –For n ≥ 3: Proceeding as in the proofof [7, Theorem 3.1], one can find a similar relation to the previous one, but connecting nEn−1 − (Ln − 4Ln−1)= n Stirling numbers of the first type and a particular class of generalized Euler polynomials. ∑ (kEk−1 − 3(k − 1)Ek−2)Ln−k+1 k=3 N n k−2 Theorem 9.Let us assume that α = m ∈ . Then, the k−s k−s−2 connection between the Stirling numbers of the first kind +5 (−1) 2 sEs−1Ln−k+1, ∑ ∑ (m) k=3 s=1 and the generalized Euler polynomial En (x) is given by n−1 the formula: (−1) nEn−1 − Ln − (α − 2x − 3)Ln−1 = n 1 n n n−k (−1)k−1 (kE + 3(k − 1)E )L E(m)(x)= s(n − k, j)(−m) j (2x)k, ∑ k−1 k−2 n−k+1 n 2n ∑ k ∑ k=3 k=0 " j=0 # n k−2 (41) k−1 k−s−2 +5 ∑ ∑ (−1) 2 sLn−k+1Es−1. k=3 s=1 Proof.By Leibniz’s theorem for differentiation, we have Other similar combinatorial identities may be obtained ∂ r 2 m using the results of [14]. We leave their formulation to the exz ∂zr ez + 1 interested reader. r r 2 m (k) ∂ r−k = (exz) ∑ k ez + 1 ∂zr−k 5 Euler matrices and their relation with k=0 Stirling and Vandermonde matrices r r 2 m (k) = xr−kexz ∑ k ez + 1 Let s(n,k) and S(n,k) be the Stirling numbers of the first k=0 m r r−k and second types, which are respectively defined by the 2 r (−m)k x = e(x+1)z , generating functions [21, Chapther 1, Section 1.6]: ez + 1 ∑ k (ez + 1)k k=0 n where in the last expression, (−m) denotes the falling k k ∑ s(n,k)z = z(z − 1)···(z − n + 1), (39) factorial with opposite argument −m. k=0 Combining this with the r-th differentiation on both ∞ zn sides of the generating function in (1) reveals that (log(1 + z))k = k! s(n,k) , |z| < 1, ∑ n! n=k ∞ zn−r n E(m)(x) = n ∑ n n r ! z = ∑ S(n,k)z(z − 1)···(z − k + 1), n=r ( − ) k=0 2 m r r (−m) xr−k ∞ n e(x+1)z k . z k z ez + 1 ∑ k (ez + 1)k (e − 1) = k! ∑ S(n,k) . k=0 n=k n! Further taking z → 0 and employing (39) give The value |s(n,k)| represents the number of permutations of n elements with k disjoint cycles. While, r r xr−k E(m)(x)= (−m) the Stirling numbers of the second type S(n,k) give the r ∑ k k 2k number of partitions of n objects into k non-empty k=0 r subsets. Another way to compute these numbers is by 1 r k = (−m) (2x) means of the formula (see [6, Eq. (5.1)] or [19, p. 226]): 2r ∑ k r−k k=0 k r r−k 1 k 1 r j k S(n,k)= (−1)k−l ln, 1 ≤ k ≤ n. = ∑ ∑ s(r − k, j)(−m) (2x) . k! ∑ l 2r k l=0 k=0 " j=0 #

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Finally, changing r by n, the proof of the formula (41) Remark.Notice that using (40) and putting m = 1 into (42), is completed. we get i E (x)=A(1)(x) j i− j i, j Definition 5.For the Stirling numbers s(i, j) and S(i, j) of the first and the second types, respectively, define S and i i− j i − j S =(−1)i− j (−1)k to be the (n + 1) × (n + 1) matrices by j ∑ k k=0 i− j−k r Si, j = s(i, j), i ≥ j, 0, otherwise, and 1 × − r!S(i − j − k + 1,r + 1) xk. ∑ 2 " r=0 # S(i, j), i ≥ j, S = i, j 0, otherwise. Definition 6.The (n + 1) × (n + 1) shifted Euler polynomial matrix E˜(x) is given by The matrices S and S are called Stirling matrix of the first and the second types, respectively (see [3]). E˜i, j(x)= Ei( j + x), 0 ≤ i, j ≤ n.

To obtain factorizations for Euler matrices via Stirling Let us consider the Vandermonde matrix: matrices of the first type, we will need the following 11 1 ··· 1 N S(m) matrix: For m ∈ , let be the (n + 1) × (n + 1) x 1 + x 2 + x ··· n + x matrix whose (i, j)-entries are defined by  2 2 2 2 V (x) := x (1 + x) (2 + x) ··· (n + x) ...... i i− j  . . . . .  ∑ s(i − j,k)(−m)k, i ≥ j,  . . . . .  (m) j k=0  n n n n S = x (1 + x) (2 + x) ··· (n + x)  i, j     0, otherwise. In [4, Theorem 2.1], the following factorization for the Vandermonde matrix V (x) was stated. Next theorem shows the corresponding factorizations V ˜ T of the generalized Euler matrix E (m), m ∈ N, in terms of (x) = ([1] ⊕ Sn+1)∆n+1(x)P T Stirling matrices of the first type, when the expression (41) := ([1] ⊕ Sñ+1)∆n+1(x)(P[1]) , (43) is incorporated. where Sñ+1 is the factorial Stirling matrix, i.e. the (n + 1) × (n + 1) matrix whose (i, j)-th entry is given by N Theorem 10.For m ∈ , the generalized Euler matrix S˜i, j,n+1 := j!Si, j, i ≥ j and otherwise 0, and E (m) x can be factorized, as follows: T ( ) ∆n+1(x)(P[1]) represents the LU-factorization of a lower triangular matrix whose (i, j)-th entry is x , if i ≥ j and (m) (m) i− j E (x)= S P[x]. (42) otherwise 0. The relation between the shifted Euler polynomial (m) E˜ V ˜ Proof.For m ∈ N and i ≥ j, let A (x) be the (i, j)-th entry matrix (x) and the matrices (x) and Sn+1 is contained i, j in the following result. of the matrix product S(m)P[x], then Theorem 11.The shifted Euler polynomial matrix E˜(x) i (m) (m) can be factorized in terms of the Vandermonde matrix A x S p x i, j ( ) = ∑ i,k k, j( ) V k= j (x), and thereby in terms of the factorial Stirling matrix Sñ+1, as follows: i i k i−k = s(i − k,r)(−m)r xk− j ∑ k j ∑ E˜(x)= EV (x), (44) k= j "r=0 # i i i − j i−k = 2 j−i s(i − k,r)(−m)r (2x)k− j ˜ ˜ T ∑ j k − j ∑ E (x)= E ([1] ⊕ Sn+1)∆n+1(x)P . (45) k= j "r=0 # i i− j i− j−k E˜ j i − j Proof.Let i, j(x) be the (i, j)-th entry of the shifted Euler = s(i − j − k,r)(−m)r (2x)k 2i− j ∑ k ∑ polynomial matrix E˜(x). Then, using (9), we get k=0 " r=0 # i (m) i i i = Ei j (x), k j − E˜i, j(x)= Ei( j+x)= Ei k( j+x) = Ei,kVk, j(x). ∑ k − ∑ k=0 k=0 The last equality is an immediate consequence of (41), Hence, (44) follows from this chain of equalities. The and (42) follows from the previous chain of equalities. relation (45) is a straightforward consequence of (43).

c 2020 NSP Natural Sciences Publishing Cor. 12 Y. Quintana et al.: Euler matrices and their algebraic properties revisited

Remark.Note that the relations (44) and (45) are the with each one of these q-operations (see [6, Definition analogous of [32, Eqs. (37), (38)], respectively, in the 25]). context of Euler polynomial matrices. For instance, the version of the generalized Pascal matrix of first type in this context was introduced in [34], In the present paper, all matrix identities have been as follows: Let x be any nonzero real number. The expressed using finite matrices. Since such matrix q-Pascal matrix P[x] is an (n + 1) × (n + 1) matrix whose identities involve lower triangular matrices, they have a (i, j)-entries are given by: resemblance for infinite matrices. For instance, let 1 i i− j E∞ x + , P∞[x] and E∞ be the infinite cases of the x , i ≥ j, 2 j q matrices E x + 1 , P[x] and E, respectively. Then, the p (x)= 2 i, j,q  following identity holds 0, otherwise,  1 i  E x + = P [x]E . where j denotes the q-binomial coefficient. ∞ 2 ∞ ∞ q In [11,34], some properties of the q-Pascal matrix are shown, for example, its matrix factorization by special We leave the formulation of the other analogous summation matrices, the factorization of its inverse identities to the reader. matrix, and its relation with the q-analogue of the Finally, in addition to Theorem 11, we would like to generalized Bernoulli matrix. state that an interesting application in connection with the The q-Euler polynomials En,q(x) and the generalized subjects treated in the present paper is its use in the (α) LU-factorization of certain class of matrices, whose q-Euler polynomials En,q (x) of (real or complex) order α, components can be represented by a Wronskian of some are defined, as follows (see [11,12]): function set (cf., for instance [30] for the case of α ∞ n 2 (α) z generalized Pascal functional matrices). eq(xz)= En,q (x) , (46) e (z)+ 1 ∑ [n] ! q n=0 q α 6 Perspective where |z| < π, 1 := 1, and E (x) := E(1)(x), n ∈ N . (47) We have showed some new algebraic properties of the n,q n 0 α generalized Euler polynomial matrix E ( )(x) and the (α) (α) As usual, the numbers E := E (0) and Euler matrix E . Taking into account some properties of n,q n,q E := E (0) are called the generalized q-Euler Euler polynomials and numbers, we have deduced n,q n,q numbers of order α and the generalized q-Euler numbers, E (α) product formulae for (x) and defined the inverse respectively. We introduce the q-analogue of the classical E matrix of . Also, we have established some explicit Euler numbers by means of the following generating E expressions for the Euler polynomial matrix (x), which function involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. 2 ∞ zn = ∑ εn,q . (48) The paper [11] addressed an explicit formula to the eq(z)+ eq(−z) [n]q! inverse matrix of the q-Pascal matrix plus one in terms of n=0 the q-analogue of the Euler matrix E . Thus, combining From (47) and (48), it is possible to check that if 1 some results of the present paper helps define the inverse commutes with −1, the connection between the matrix of the q-analogue of the specialized Euler matrix q-analogue of the classical Euler numbers and the q-Euler E. This inverse matrix allows us to establish the polynomials is given by the formula analogous of Theorem 3 in the framework of the class of generalized q-Pascal matrices. 1 ε = 2nE , n ∈ N . A future work has to address the exploration of n,q n,q 2 0 similar matrix identities in the setting of the q-umbral calculus. The notion of commutativity with respect to the Finally: R parameter q appears when is endowed with some (a)The q-Euler polynomials E (x) are different from structure of alphabet and its elements are called letters or n,q R those defined by Srivastava and Choi in [21, Section umbrae (see e.g. [6, Section 4.1], for the details). If is 6.7]. regarded as a set generated by itself together with the (b)For α = m ∈ N, the generalized q-Euler polynomials operations of NWA q-addition, NWA q-subtraction, JHC (α) q-addition and JHC q-subtraction, then the algebraic En,q (x) coincide with the so-called structure of R as such an alphabet is different from the Nalli-Ward-Al-Salam q-Euler polynomials (in short, (m) usual algebraic structure of R as real number field. NWA q-Euler polynomials), FNWA,n,q(x) (see for Moreover, R has structure of commutative semigroup instance, [6, Eq. (4.196)]).

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(c)From the generating relation (46), it is fairly [11] G. I. Infante, J. L. Ram´ırez and A. S¸ahin, Some results on straightforward to deduce the addition formula: q-analogue of the Bernoulli, Euler and Fibonacci matrices, Math. Rep. (Bucur.), 19(69), 399-417 (2017). n n [12] D. S. Kim and T. Kim, Some identities of q-Euler E(α+β ) x y E(α) x E(β ) y , (49) n,q ( + )= ∑ k,q ( ) n−k,q( ) polynomials arising from q-umbral calculus, J. Inequal. k k q =0 Appl., 2014:1, 1-12 (2014). provided that x commutes with y (cf., [6, Eq. (4.7)]). [13] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and (d)Making the substitution β = 0 into (49) and eigenvalues of Fibonacci and symmetric Fibonacci matrices, interchanging x and y, we get Fibonacci Quart., 40, 203-211 (2002). [14] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial n n identities via Fibonacci numbers, Discrete Appl. Math., 130, E(α)(x + y)= E(α)(y)xn−k. n,q ∑ k k,q 527-534 (2003). k=0 q [15] Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308, 290-302 (2005). Acknowledgement [16] N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin DE, 120-162, (1924). The first author (YQ) has been supported by the grants [17] A.´ Pintér and H. M. Srivastava, Addition theorems for the Impacto Caribe (IC-002627-2015) from Universidad del Appell polynomials and the associated classes of polynomial Atlántico, Colombia, and DID-USB (S1-IC-CB-003-16) expansions, Aequationes Math., 85, 483-495 (2013). from Decanato de Investigación y Desarrollo. [18] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational Universidad Simón Bol´ıvar, Venezuela. matrix method based on generalized Bernoulli polynomials of level m, Calcolo 55, 1-29 (2018). The second (WR) and third (AU) authors have been [19] J. 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c 2020 NSP Natural Sciences Publishing Cor. 14 Y. Quintana et al.: Euler matrices and their algebraic properties revisited

[31] Z. Z. Zhang and M. X. Liu, An extension of generalized William Ram´ırez Pascal matrix and its algebraic properties, Linear Algebra is Professor and researcher in Appl., 271, 169-177 (1998). Faculty of Natural and Exact [32] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic Sciences at Universidad properties, Discrete Appl. Math., 154, 1622-1632 (2006). de la Costa (Colombia), [33] Z. Zhang and Y. Zhang, The Lucas matrix and some and previously was member combinatorial identities, Indian J. Pure Appl. Math., 38, 457- of Department of Pure 465 (2007). Mathematics at Universidad [34] D. Zheng, q-analogue of the Pascal matrix, Ars Combin., 87, del Atlántico (Colombia). He 321-336 (2008). received the MSc degree in Mathematical Sciences at Universidad del Atlántico. Also, he is a junior researcher recognized by Colciencias Yamilet Quintana (Colombia). His main research interests are: mathematical is currently Full Professor analysis, number theory and special functions. in Department of Pure and Applied Mathematics at Universidad Simón Bol´ıvar (Venezuela). She received Alejandro Urieles her PhD in Mathematics is Professor and from Universidad Central researcher in Faculty of de Venezuela in 2004. Basic Sciences-Mathematics Her research interest includes Program at Universidad approximation theory and special functions, mathematical del Atlántico (Colombia), analysis, harmonic analysis for orthogonal polynomials and previously was Assistant expansions, didactic aspects of mathematical analysis and Professor in Department didactic aspects of geometry. She has published research of Pure and Applied articles in reputed international journals of of pure and Mathematics at Universidad applied mathematics. She serves as both a referee and Simón Bol´ıvar (Venezuela). Editorial Board member for several international journals. He received the PhD degree in Mathematics at Universidad Simón Bol´ıvar. He is a senior researcher recognized by Colciencias (Colombia) and referee of international journals in the frame of pure and applied mathematics. His research has been based mainly on approximation theory and special functions, mathematical analysis and harmonic analysis for orthogonal polynomials expansions. Currently, he coordinates the Hotbed of Research on Approximation Theory and Special Functions at Universidad del Atlántico.

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