SOME RESULTS on G-ANALOGUE of the BERNOULLI, EULER AND

SOME RESULTS ON q-ANALOGUE OF THE BERNOULLI, EULER AND FIBONACCI MATRICES

GERALDINE M. INFANTE, JOSE´ L. RAM´IREZ and ADEM S¸AHIN˙

Communicated by Alexandru Zaharescu

In this article, we study q-analogues of the Bernoulli, Euler and Fibonacci matrices. Some algebraic properties of these matrices are presented and proved. In particular, we show various factorizations of these matrices and their inverse matrices. AMS 2010 Subject Classiﬁcation: 15A24, 11B68, 15A09, 15A15. Key words: q-Bernoulli matrix, q-Euler matrix, q-Fibonacci matrix, q-analogues, combinatorial identities.

1. INTRODUCTION

Matrix theory is recently used in the study of several combinatorial sequences. The matrix representation gives a powerful tool to obtain new or classical identities. In particular, Pascal type matrices [5,7,26,36] have been used to obtain new interesting combinatorial identities involving sequences such as Fibonacci and Lucas sequence [22,23,32,37], k-Fibonacci numbers [21], Catalan numbers [31], Stirling numbers and their generalizations [8,11,22,24,25,27,28], Bernoulli numbers [8,35], among others. In the present paper, we are interested in lower-triangular matrices whose entries are q-Bernoulli numbers [1, 19], q- Euler numbers [20] and q-Fibonacci numbers [12]; for more details about these sequences see [13]. The q-Bernoulli polynomials, Bn,q(x), are deﬁned by the exponential generating function ∞ n teq(xt) X t F (x, t, q) := = B (x) (|t| < 2π), e (t) − 1 n,q [n] ! q n=0 q where the q-exponential function eq(x) is deﬁned by ∞ X xn 1 e (x) := = (0 < |q| < 1, x < |1 − q|−1), q [n] ! ((1 − q)x; q) n=0 q ∞

MATH. REPORTS 19(69), 4 (2017), 399–417 400 Geraldine M. Infante, Jos´eL. Ram´ırez and Adem S¸ahin 2 with ∞ Y j n−1 (a; q)∞ := (1−aq ), [n]q = 1+q+···+q and [n]q! = [1]q[2]q ··· [n]q. j=0

In particular, if x = 0 in Bn,q(x), we obtain the q-Bernoulli numbers Bn,q (cf. [13]). The q-exponential function satisfies the addition formula eq(x+y) = eq(x)eq(y) provided that yx = qxy (q-commutative variables). For a real or complex parameter α, the generalized q-Bernoulli polyno- (α) mials Bn,q (x) are defined by the generating function (1) α ∞ t X tn G(x, t, α, q) := e (xt) = B(α)(x) (|t| < 2π, α ∈ ). e (t) − 1 q n,q [n] ! C q n=0 q (1) (1) Clearly, Bn,q(x) = Bn,q(x) and Bn,q(0) = Bn,q are the q-analogue of the Ber- noulli polynomials and numbers, respectively. The q-binomial coefficient is defined as n (q; q) := n , k q (q; q)k(q; q)n−k and n−1 Y j (a; q)n := (1 − aq ). j=0 Another way to write the q-binomial coefficient is n [n] ! = q . k q [k]q![n − k]q! From (1) we can derive the following equation n X n (α) (β) (2) B(α+β)(x + y) = B (x)B (y) n,q k k,q n−k,q k=0 q provided that x commutes with y, i.e., if yx = qxy. Indeed, ∞ i α+β X (α+β) t t B (x + y) = e ((x + y)t) i,q [i] ! e (t) − 1 q i=0 q q t α t β = eq(xt) eq(yt) eq(t) − 1 eq(t − 1 ∞ i ∞ i X (α) t X (β) t = B (x) B (y) i,q [i] ! i,q [i] ! i=0 q i=0 q 3 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 401

∞ i ! i X X i (α) (β) t = Bl,q (x)Bi−l,q(y) . l [i]q! i=0 l=0 q The result follows by comparing the coefficients. In particular, upon setting β = 0 in identity (2) and interchanging x and y, we obtain n X n (α) B(α)(x + y) = B (y)xn−k. n,q k k,q k=0 q Specifically, if α = 1 n X n (3) B (x + y) = B (y)xn−k, n,q k k,q k=0 q and if y = 0 n X n B (x) = B xk. n,q k n−k,q k=0 q The first few q-Bernoulli polynomials are 1 q2 B (x) = 1,B (x) = x− ,B (x) = x2 −x+ , 0,q 1,q q + 1 2,q (q + 1) (q2 + q + 1) q2 + q + 1 q2 (q − 1)q3 B (x) = x3 − x2 + x − , 3,q q + 1 q + 1 (q + 1)2 (q2 + 1) q2 q2 + 1 q3 − q4 B (x) = x4 − q2 + 1 x3 + x2 + x 4,q q + 1 q + 1 q10 − q8 − 2q7 − q6 + q4 + . (q + 1)2 (q2 + q + 1) (q4 + q3 + q2 + q + 1)

The q-Euler polynomials En,q(x) are deﬁned by means of the following generating function (cf. [20]) ∞ X tn 2 E (x) := e (xt)(|t| < π). n,q [n] ! e (t) + 1 q n=0 q q

In the special case, x = 0, En,q(0) := En,q are called the q-Euler numbers. The first few q-Euler numbers are 1 1 q − 1 1 E = 1,E = − ,¨ı?‘ E = ,E = − (q+1) q2 − 3q + 1, 0,q 1,q 2 2 2,q 4 3,q 8 1 E = (q − 1)(q + 1) q2 − 4q + 1 q2 + q + 1 , ··· . 4,q 16 On the other hand, there exist several slightly different q-analogues of the Fibonacci sequence, among other references, see, [2–4, 9, 12, 29]. In particular, 402 Geraldine M. Infante, JoséL. Ram´ırez and Adem S¸ahin 4 we are interested in the polynomials introduced by Cigler [12]. The q-Fibonacci polynomials, Fn,q(x, s), are defined as follows

b n−1 c 2 X n − k − 1 k+1 n−1−2k k F (x, s) := q( 2 )x s . n,q k k=0 q

In particular, if we take x = s = 1, we obtain the q-Fibonacci numbers Fn,q. The ﬁrst few q-Fibonacci number are

2 F0,q = 0,F1,q = 1,F2,q = 1,F3,q = q + 1,F4,q = q + q + 1, 3 2 5 4 3 2 F5,q = 2q + q + q + 1,F6,q = q + 2q + 2q + q + q + 1, 7 6 5 4 3 2 F7,q = q + 2q + 3q + 2q + 2q + q + q + 1,... It is clear that if q = 1 we recover the well-known Fibonacci sequence. In the present article, we study three families of matrices, the q-Bernoulli matrix, the q-Euler matrix and the q-Fibonacci matrix. Then we obtain several results that generalize the classical case (q = 1). The outline of this paper is as follows. In Section 2, we introduce the generalize q-Bernoulli matrix, then we derive some basic identities, in particular we find its inverse matrix. In Section 3, we find a factorization of the q-Bernoulli matrix in terms of the q-Pascal matrix. In Section 4, we show a relation between the q-Pascal matrix plus one and the q-Euler matrix. Finally, in Section 5 we introduce the q-Fibonacci matrix and we find its inverse matrix and some interesting factorizations. In particular, we show a relation between the q-Bernoulli polynomial matrix and the q-Fibonacci matrix.

2. THE q-ANALOGUE OF THE GENERALIZED BERNOULLI MATRIX

We deﬁne the q-analogue of the generalized (n + 1) × (n + 1) Bernoulli (α) (α) (α) matrix Bn,q (x) := Bq (x) := [Bi,j (x)](0 ≤ i, j ≤ n), where

( (α) i B (x), if i ≥ j; (α) j q i−j,q Bi,j (x) = 0, otherwise.

(1) (1) (1) The matrices Bn,q(x) := Bq (x) := Bq(x) and Bn,q(0) := Bq(0) := Bq are called the q-Bernoulli polynomial matrix and the q-Bernoulli matrix, respectively. For more information about this matrix see [13, 16]. Note that this matrix is diﬀerent from one recently studied by Tuglu and Ku¸s,[33]. In particular, if q = 1 we recover the generalized Bernoulli matrix [35]. 5 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 403

Example 1. For n = 3, we have

B3,q(x)  1 0 0 0  x − 1 1 0 0  q+1  = 2 q2   x − x + q3+2q2+2q+1 (q + 1)x − 1 1 0  2 3  3 (q +q+1) 2 q2 (q−1)q 2 2 q2 2 q2+q+1 x − q+1 x + q+1 x− (q+1)2(q2+1) (q +q+1)(x−x )+ 1+q (q +q+1)x− q+1 1 and  1 0 0 0 1  − q+1 1 0 0  2  B3,q =  q −1 1 0 .  q3+2q2+2q+1  (q−1)q3 q2 q2+q+1 − (q+1)2(q2+1) q+1 − q+1 1 Theorem 2. If x commutes with y, then the following equalities hold for any α and β (α+β) (α) (β) (β) (α) (4) Bq (x + y) = Bq (x)Bq (y) = Bq (y)Bq (x). We need the following q-binomial identity. Lemma 3. The following identity holds for any positive integers i, j, k i k i i − j = . k q j q j q k − j q Proof. From the definition of q-binomial coefficient we get i i − j [i] ! [i − j] ! = q q j q k − j q [j]q![i − j]q! [k − j]q![i − j − k + j]q! [i]q![k]q! [i]q! [k]q! i k = = = . [j]q![k − j]q![i − k]q![k]q! [k]q![i − k]q! [j]q![k − j]q! k q j q Proof of Theorem 2. From above lemma and Identity (2) we have i (α) (β) X i (α) k (β) (B (x)B (y))i,j = B (x) B (y) q q k i−k,q j k−j,q k=j q q i i−j X i i − j (α) (β) i X i − j (α) (β) = B (x)B (y) = B (x)B (y) j k − j i−k,q k−j,q j k i−j−k,q k,q k=j q q q k=0 q i (α+β) (α+β) = Bi−j,q (x + y) = (Bq (x + y))i,j. j q Then Equation (4) follows. 404 Geraldine M. Infante, JoséL. Ram´ırez and Adem S¸ahin 6

If q = 1 we obtain Theorem 2.1 of [35]. Corollary 4. If the variables commute then the following equality holds for any integer k ≥ 2

(α1+α2+···+αk) (α1) (α2) (αk) (5) Bq (x1 + x2 + ··· + xk) = Bq (x1)Bq (x2) ··· Bq (xk). Proof. We proceed by induction on k. From Theorem 2 the equality clearly holds for k = 2 . Now suppose that the result is true for all i < k. We prove it for k.

(α1+α2+···+αk) Bq (x1 + x2 + ··· + (xk−1 + xk))

(α1) (α2) (αk−1+αk) = Bq (x1)Bq (x2) ··· Bq (xk−1 + xk). By Theorem 2, we get

(α1) (α2) (αk−1+αk) Bq (x1)Bq (x2) ··· Bq (xk−1 + xk)

(α1) (α2) (αk−1) (αk) = Bq (x1)Bq (x2) ··· Bq (xk−1)B (xk).

If we take x1 = x2 = ··· = xk = x and α1 = α2 = ··· = αk = α , then (α) k (kα) k (k) (Bq (x)) = Bq (kx). In particular, if α = 1, then (Bq(x)) = Bq (kx); if (α) k (kα) k (k) x = 0, (Bq ) = Bq . Finally, if α = 1, x = 0 we obtain Bq = Bq , i.e., a formula of the powers of the q-Bernoulli matrix. Now, we study the inverse matrix of the q-Bernoulli matrix. Let Dq = [di,j](0 ≤ i, j ≤ n) be the (n + 1) × (n + 1) matrix deﬁned by ( 1 i , if i ≥ j; [i−j+1]q j q di,j = 0, otherwise. We need the following lemma [20]. Lemma 5. For any positive integer n n X 1 n Bn−k,q = [n]q!δn,0, [k + 1]q k k=0 q where δn,m is the Kronecker delta symbol.

Theorem 6. The inverse matrix of the q-Bernoulli matrix Bq is the ma- −1 (k) −1 k trix Dq, i.e, Bq = D. Furthermore, (Bq ) = Dq . Proof. From above lemma, we get

i X i 1 k (BqDq)i,j = Bi−k,q · k [k − j + 1]q j k=j q q 7 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 405

i i X 1 i − j = Bi−k,q j [k − j + 1]q k − j q k=j q i−j i X 1 i − j i = Bi−j−k,q = [n]q!δi−j,0. j [k + 1]q k j q k=0 q q

This is 1 only when i = j, and 0 otherwise. Therefore BqDq = I, i.e., −1 Bq = Dq. Moreover, if α = 1 and x = 0 we get

(k) −1 k −1 −1 k k (Bq ) = (Bq ) = (Bq ) = Dq . If q = 1 we obtain Theorem 2.4 of [35].

3. THE q-ANALOGUE OF THE GENERALIZED BERNOULLI MATRIX AND THE GENERALIZED q-PASCAL MATRIX

In this section, we show some relations between the q-analogues of the generalized Bernoulli matrix and the generalized Pascal matrix. The Pascal matrix is one of the most important matrices of mathematics. It arises in many different areas such as combinatorics, number theory, etc. Many kinds of generalizations of the Pascal matrix have been presented in the literature. In particular, we are interested in its q-analogue; see, e.g., [13–17, 38]. The (n + 1) × (n + 1) q-Pascal matrix Pn := Pn,q := [pi,j] (0 ≤ i, j ≤ n) is defined with the q-binomial coefficients as follows

(i , if i j; j q > pi,j := 0, otherwise.

For any nonnegative integers n and k, the (n + 1) × (n + 1) matrices (k) (k) (k) In, Sn , Dn and Pn are deﬁned by

(In)i,j := diag(1, 1,..., 1), ( (i−j)k (k) q , if i ≥ j; (S )i,j := (0 ≤ i, j ≤ n), n 0, otherwise.  1, if i = j;  (k) k (Dn )i,j = −q , if j = i − 1; 0, otherwise. (k) (i−j)k (Pn )i,j := (Pn)i,jq . 406 Geraldine M. Infante, Jos´eL. Ram´ırez and Adem S¸ahin 8

(0) Clearly, Pn = Pn. Furthermore we need the (n + 1) × (n + 1) matrices " # (k) 1 0 Pn := (k) 0 Pn−1 " # In−k−1 0 (0) Fk := (n−k) , (1 ≤ k ≤ n − 1) and Fn := Dn , 0 Dk " # In−k−1 0 (0) Gk := (n−k) , (1 ≤ k ≤ n − 1) and Gn := Sn . 0 Sk (k) −1 (k) −1 It is not diﬃcult to see that (Dn ) = Sn and Fk = Gk. Moreover, for any nonnegative integer k, we have the following factorization [38] (k) (k) (k+1) Dn Pn = Pn .

By the above equation and the deﬁnition of the matrix Fk, we get −1 −1 −1 F1F2 ···FnPn = In or Pn = Fn Fn−1 ···F1 .

Therefore, the q-Pascal matrix Pn can be factorized as follows

Pn = GnGn−1 ···G1. Moreover, it is clear that the inverse of the q-Pascal matrix can be factorized as −1 Pn = F1F2 ···Fn = Pn, i−j (i−j) where (Pn)i,j := (−1) (Pn)i,jq 2 . The (n+1)×(n+1) generalized q-Pascal matrix Pn[x] := [pi,j] (0 ≤ i, j ≤ n), is deﬁned by [38] (i xi−j, if i ≥ j; j q pi,j = 0, otherwise. Example 7. For n = 4, the generalized q-Pascal matrix is  1 0 0 0 0  x 1 0 0 0  2  P4[x] = x (q + 1)x 1 0 0   x3 (q2 + q + 1)x2 (q2 + q + 1)x 1 0 x4 (1 + q)(1 + q2)x3 (1 + q2)(1 + q + q2)x2 (1 + q)(1 + q2)x 1 In [17], Ernst studied a generalization of this matrix. Theorem 8. If x commutes with y, then we have the following identity

Bq(x + y) = Pn[x]Bq(y) = Pn[y]Bq(x).

In particular, Bq(x) = Pn[x]Bq. 9 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 407

Proof. From Equation (3), we have i X i i−k k (Pn[x]Bq(y))i,j = x B (y) k j k−j,q k=j q q i i−j i X i − j i X i − j = B (y)xi−k = B (y)xi−j−k j k − j k−j,q j k k,q q k=j q q k=0 q i = Bi−j,q(x + y) = (Bq(x + y))i,j. j q

Similarly, we have Bq(x + y) = Pn[y]Bq(x). In particular, if q = 1 we obtain Theorem 3.1 of [35]. Example 9. If n = 2 we obtain the following factorization  1 0 0  x − 1 1 0 B2,q(x) =  q+1  2 q2 x − x + q3+2q2+2q+1 (q + 1)x − 1 1    1 0 0 1 0 0  − 1 1 0 =  x 1 0 ×  q+1  = P2[x]B2,q. x2 (q + 1)x 1 q2 q3+2q2+2q+1 −1 1 (k) (k) Let Sn [x] and Dn [x] be the (n + 1) × (n + 1) matrices defined by (k) (k) i−j (k) (k) i−j (Sn [x])i,j = (Sn )i,jx and (Dn [x])i,j = (Dn )i,jx . For 1 ≤ k ≤ n − 1, we define the following matrices " # In−k−1 0 (0) Fk[x] := (k) , (1 ≤ k ≤ n − 1) and Fn[x] := Dn [x], 0 Dn−k[x] " # In−k−1 0 (0) Gk[x] := (k) , (1 ≤ k ≤ n − 1) and Gn[x] := Sn [x]. 0 Sn−k[x] −1 Clearly, Fk[x] = Gk [x], 1 ≤ k ≤ n. We need the (n + 1) × (n + 1) matrix n In[x] := diag(1, x, . . . , x ). By definition, we have [38] (k) (k) −1 Sn [x] = In[x]Sn In [x], −1 Gk[x] = In[x]GkIn [x], −1 Pn[x] = In[x]PnIn [x]. 408 Geraldine M. Infante, JoséL. Ram´ırez and Adem S¸ahin 10

Moreover, the generalized q-Pascal matrix Pn[x] can be factorized by the matrices Gk[x] as follows −1 Pn[x] = In[x]GnGn−1 ···G1In [x] −1 −1 −1 = (In[x]GnIn [x])(In[x]Gn−1In [x]) ··· (In[x]G1In [x]) = Gn[x]Gn−1[x] ···G1[x]. Moreover, −1 Pn [x] = Fn[x]Fn−1[x] ···F1[x] = Pn[x], (i−j) i−j where (Pn[x])i,j := (Pn)i,jq 2 (−x) . From above factorizations we have the following theorem. Theorem 10. For any x we have the following identities

Bq(x) = Gn[x]Gn−1[x] ···G1[x]Bq, −1 Bq(x) = DqFn[x]Fn−1[x] ···F1[x]. If q = 1 we obtain Corollary 3.3 of [35].

4. INVERSE MATRIX OF THE q-PASCAL MATRIX PLUS ONE

In this section, we give an explicit formula to the inverse matrix of the q-Pascal matrix plus one, and we show the relation between this inverse matrix and the q-Euler numbers. Yang and Liu [34] studied the case q = 1. Lemma 11. For any n ≥ 0, we have n 1 X n 1 E (x) + E (x) = xn, 2 k k,q 2 n,q k=0 q where En,q(x) are the q-Euler polynomials defined by ∞ X tn 2 E (x) = e (xt), (|t| < π). n,q [n] ! e (t) + 1 q n=0 q q Proof. From definition of q-Euler numbers we get ∞ ! X tn 2 E (x) (e (t) + 1) = e (xt)(e (t) + 1) = 2e (xt). n,q [n] ! q e (t) + 1 q q q n=0 q q Therefore, ∞ n ! ∞ X X n tn X tn E (x) + E (x) = 2xn . i i,q n,q [n] ! [n] ! n=0 i=0 q q n=0 q By comparing coefficients, we get the desired result. 11 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 409

Lemma 12. If x commutes with y, then for any n ≥ 0, we have n X n E (x + y) = E (x)yn−i. n,q i i,q i=0 q In particular, if x = 0 we have n X n E (y) = E yn−i. n,q i i,q i=0 q Proof. From deﬁnition of q-Euler numbers and since x, y are q-commutative variables, we get ∞ X tn 2 2 E (x + y) = e ((x + y)t) = e (xt)e (yt) n,q [n] ! e (t) + 1 q e (t) + 1 q q n=0 q q q ∞ ! ∞ n ! X tn X X n tn = E (x) e (yt) = E (x)yn−i . n,q [n] ! q i i,q [n] ! n=0 q n=0 i=0 q q By comparing coeﬃcients, we get the desired result.

We deﬁne the q-analogue of the (n + 1) × (n + 1) Euler matrix En,q := [Ei,j](0 ≤ i, j ≤ n), where

(i Ei−j,q, if i ≥ j; j q Ei,j = 0, otherwise. Theorem 13. For all positive integer n the following identity holds

−1 1 (Pn + In) = En,q. 2 Proof. From Lemma 11 we have the following matrix equation 1 (P + I ) (x) = , 2 n n En,q Xn where En,q(x) and Xn are n × 1 matrices deﬁned by T n−1T En,q(x) = [E0,q(x),E1,q(x),...,En−1,q(x)] , Xn = 1, x, . . . , x . From Lemma 12 we have the following matrix equation

En,q(x) = En,qXn. Therefore 1 (Pn + In)En,q = In. 2 If q = 1 we obtain the main result in [34]. 410 Geraldine M. Infante, Jos´eL. Ram´ırez and Adem S¸ahin 12

5. SOME RELATIONS BETWEEN q-PASCAL MATRIX AND q-FIBONACCI MATRIX

In this section, we give some relations between q-Pascal matrix and q- Fibonacci matrix. Moreover, we obtain the inverse matrix of the q-Fibonacci matrix. We deﬁne the q-analogue of the (n + 1) × (n + 1) Fibonacci matrix Fn,q := Fq := [fi,j](0 ≤ i, j ≤ n), where ( Fi−j,q, if i ≥ j; fi,j = 0, otherwise. A generalization of the q-Fibonacci matrix and its inverse were recently studied in [30]. Example 14. The q-Fibonacci matrix for n = 4 is  1 0 0 0 0  1 1 0 0 0   F4,q =  q + 1 1 1 0 0 .    q2 + q + 1 q + 1 1 1 0 2q3 + q2 + q + 1 q2 + q + 1 q + 1 1 1

We need the following auxiliary sequence and matrix. Let Nn be the sequence deﬁned by N0 = 1, N1 = 1 and n−1 X n−k Nn = Nn−1 + (−1) Fn−k+2,qNk−1, n ≥ 2. k=1

The ﬁrst few terms of Nn are 1, 1, −q, q2−q, 2q2−2q3, q5+2q4−4q3+q2, −q7−2q6−q5+7q4−3q3,....

The (n + 1) × (n + 1) lower Hessenberg matrix Cn := [ci,j](0 ≤ i, j ≤ n) is deﬁned by ( Fi−j+2,q, i − j + 2 ≥ 1; ci,j = 0, otherwise.

Lemma 15 ([6]). Let An be an n×n lower Hessenberg matrix for all n ≥ 1 and deﬁne det(A0) = 1. Then, det(A1) = a11 and for n ≥ 2 n−1 n−1 X n−r Y det(An) = an,n det(An−1) + [(−1) an,r( aj,j+1) det(Ar−1)]. r=1 j=r

Lemma 16. For n ≥ 1, det(Cn) = Nn. 13 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 411

Proof. We proceed by induction on n. The result clearly holds for n = 1. Now suppose that the result is true for all positive integers less than or equal to n. We prove it for n + 1. In fact, by using Lemma 15 we have

n  n  X n+1−i Y det(Cn+1) = cn+1,n+1 det(Cn) + (−1) an+1,i aj,j+1 det(Ci−1) i=1 j=i n X n+1−i = det(Cn) + (−1) Fn−i+3,q det(Ci−1) i=1 n X n+1−i = Nn + (−1) Fn−i+3,qNi−1 = Nn+1. i=1 We need the following Theorem of Chen and Yu [10].

Theorem 17. Let H = [hi,j] be an n × n lower Hessenberg matrix,  1 0 ··· 0   .  ˜  .  H =    H 0  1 [α] [L] and H˜ −1 = n×1 n×n . Then h βT 1×n (n+1)×(n+1) (6) det(H) = (−1)nh · det(H˜ ),

(7) L = H−1 + h−1αβT ,

(8) Hα + hen = 0, where en is nth column of matrix In.

Lemma 18 ([18]). Let e1 be the ﬁrst column of the identity matrix In and L, β, h be the matrices described in the Theorem 17. Then, T (9) β H + he1 = 0.

Theorem 19. Let Fn,q be the (n+1)×(n+1) lower triangular q-Fibonacci matrix, then its inverse matrix is given by

 i−j (−1) Ni−j, i > j; −1  (Fn,q) = [bi,j] = 1, i = j; 0, otherwise. 412 Geraldine M. Infante, Jos´eL. Ram´ırez and Adem S¸ahin 14

Proof. It is obvious that,  1 0 ··· 0   .  ˜  .  (Cn) =   = Fn+1,q.  Cn 0  1

−1 [α] [L] Let us construct the matrix ( ˜ ) = n×1 n×n and Cn h βT 1×n (n+1)×(n+1) obtain the entries. n (−1) det(Cn) n (1) By using (6) we obtain h = = (−1) det(Cn), det(C˜n) (2) We obtain matrices [α] and [β] by using (8) and (9):  1   − det C1    −1 n  .  [α] = −(Cn) (−1) det(Cn)en =  .   n−2   (−1) det Cn−2  n−1 (−1) det Cn−1 T n −1 and [β ] = −(−1) det(Cn)e1(Cn) . Therefore T n−1 n−2 [β ] = (−1) det Cn−1 (−1) det Cn−2 · · · − det C1 1 . (3) We obtain matrix [L] by using (7) and Lemma 16:

−1 n −1 T [L] = (Cn) + ((−1) det Cn) αβ  1 0 ··· 0   .. .. .   − det C1 . . .  =   .  . .. ..   . . . 0  n−2 (−1) det Cn−2 · · · − det C1 1

Consequently, combining these three steps and using Lemma 16, we obtain the required result. In particular, if q = 1 we obtain the inverse matrix of the Fibonacci matrix [23]. Lemma 16 and Theorem 19 were recently generalized in [30].

Example 20. The inverse matrix of F4,q is  1 0 0 0 0   −1 1 0 0 0  −1   (F4,q) =  −q −1 1 0 0  .    q − q2 −q −1 1 0  2q2 − 2q3 q − q2 −q −1 1 15 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 413

The (n + 1) × (n + 1) lower triangular matrix Rn,q := [ri,j], (0 ≤ i, j ≤ n) is deﬁned by (Pi−j k i (−1) Nk, if i j; k=0 j+k q > ri,j = 0, otherwise.

Theorem 21. The following equality holds for any positive integer n

Pn,q = Rn,qFn,q.

−1 Proof. Note that it suﬃces to prove that Pn,q(Fn,q) = Rn,q. For i ≥ j ≥ 0 we have

n n i X X i X i p b = (−1)k−jN = (−1)k−jN i,k k,j k k−j k k−j k=0 k=0 q k=j q i−j X i = (−1)k N = r j + k k i,j k=0 q

−1 and it is obvious that ri,j = 0 for i − j > 0, which implies that Pn,q(Fn,q) = Rn,q, as desired. In particular, if q = 1 we obtain Theorem 2.1 of [37].

Example 22.

1 0 0 0 0 1 1 0 0 0   P4,q = 1 q + 1 1 0 0   1 q2 + q + 1 q2 + q + 1 1 0 1 q3 + q2 + q + 1 q4 + q3 + 2q2 + q + 1 q3 + q2 + q + 1 1  1 0 0 0 0  0 1 0 0 0   =  −2q q 1 0 0 ×    −q3 − 3q2 − q −q q2 + q 1 0 −2q5 − q4 − 5q3 − q −2q4 − q3 − 3q2 q4 + q2 − q q3 + q2 + q 1  1 0 0 0 0  1 1 0 0 0    q + 1 1 1 0 0 = R4,q · F4,q.    q2 + q + 1 q + 1 1 1 0 2q3 + q2 + q + 1 q2 + q + 1 q + 1 1 1 414 Geraldine M. Infante, Jos´eL. Ram´ırez and Adem S¸ahin 16

The (n + 1) × (n + 1) lower triangular matrix Sn,q := [si,j], (0 ≤ i, j ≤ n) is deﬁned by Pi−j k+1j+k  k=0(−1) j Ni−j−k, if i > j > 1, i − j odd;  q Pi−j kj+k si,j = k=0(−1) j Ni−j−k, if i > j > 1, i − j even;  q 0, otherwise. Theorem 23. The following equality holds for any positive integer n

Pn,q = Fn,qSn,q.

Proof. The proof runs like in Theorem 21. In particular, if q = 1 we obtain the Theorem 2.1 of [22]. Example 24.

1 0 0 0 0 1 1 0 0 0   P4,q = 1 q + 1 1 0 0   1 q2 + q + 1 q2 + q + 1 1 0 1 q3 + q2 + q + 1 q4 + q3 + 2q2 + q + 1 q3 + q2 + q + 1 1  1 0 0 0 0  1 1 0 0 0   =  q + 1 1 1 0 0 ×    q2 + q + 1 q + 1 1 1 0 2q3 + q2 + q + 1 q2 + q + 1 q + 1 1 1  1 0 0 0 0  0 1 0 0 0    −q q 1 0 0 = F4,q ·S4,q.    −q2 −q + q2 q + q2 1 0 q2 − 2q3 −2q2 + q3 −q + q2 + q3 + q4 q + q2 + q3 1 Finally, we show a relation between the q-Fibonacci matrix and the q- Bernoulli matrix. The (n + 1) × (n + 1) lower triangular matrix Tn,q := [ti,j], 0 ≤ i, j ≤ n is deﬁned by (i Pi−1 i−kk Bi−j,q(x) + (−1) Ni−kBk−j,q(x), if i j 0 j q k=j j q > > ti,j = 0, otherwise. Theorem 25. The following equality holds for any positive integer n

Bn,q(x) = Fn,qTn,q. 17 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 415

Proof. The proof runs like in Theorem 21. In particular, if q = 1 we obtain the Theorem 4.1 of [35]. Example 26. If x = 0 and n = 3 we get  1 0 0 0   1 1 0 0 0  − q+1 1 0 0 1 1 0 0  2    B3,q =  q −1 1 0 =    q3+2q2+2q+1   q + 1 1 1 0 (q−1)q3 q2 q2+q+1 q2 + q + 1 q + 1 1 1 − (q+1)2(q2+1) q+1 − q+1 1  1 0 0 0 q+2  − q+1 1 0 0  4 3  ×  q +2q −1  = F3,q ·T3,q.  − (q+1)(q2+q+1) −2 1 0  7 6 5 3 2  q(−q −2q −2q +3q +5q +3q+2) 1 q2+2q+2 (q+1)2(q2+1)(q2+q+1) q+1 − q+1 1

Acknowledgments. The authors thank the anonymous referee for her/his comments and remarks, which helped us to improve the article.

REFERENCES

[1] W.A. Al-Salam, q-Bernoulli numbers and polynomials. Math. Nachr. 17 (1959), 239– 260. [2] G.E. Andrews, Fibonacci numbers and the Rogers-Ramanujan identities. Fibonacci Quart. 42 (2004), 1, 3–19. [3] H. Belbachir and A. Benmezai, An alternative approach to Cigler’s q-Lucas polynomials. Appl. Math. Comput. 226 (2014), 691–698. [4] H. Belbachir and A. Benmezai, A q analogue for bisnomial coeﬃcients and generalized Fibonacci sequence. C. R. Math. Acad. Sci. Paris I 352 (2014), 3, 167–171. [5] R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix. Linear Algebra Appl. 174 (1992), 13–23. [6] N.D. Cahill, J.R. D’Errico, D.A. Narayan and J.Y. Narayan, Fibonacci determinants. College Math. J. 33 (2002), 221–225. [7] G.S. Call and D.J. Velleman, Pascal’s matrices. Amer. Math. Monthly 100 (1993), 4, 372–376. [8] M. Can and M. Cihat Da˘gli, Extended Bernoulli and Stirling matrices and related combinatorial identities. Linear Algebra Appl. 444 (2014), 114–131. [9] L. Carlitz, Fibonacci notes. III. q-Fibonacci numbers. Fibonacci Quart. 12 (1974), 317–322. [10] Y-H. Chen and C-Y. Yu, A new algorithm for computing the inverse and the determinant of a Hessenberg matrix. Appl. Math. Comput. 218 (2011), 4433–4436. [11] G.S. Cheon and J.S. Kim, Factorial Stirling matrix and related combinatorial sequences. Linear Algebra Appl. 357 (2002), 247–258. [12] J. Cigler, A new class of q-Fibonacci polynomials. Electron. J. Combin. 10 (2003), # R19. 416 Geraldine M. Infante, Jos´eL. Ram´ırez and Adem S¸ahin 18

[13] T. Ernst, A Comprehensive Treatment of q-Calculus. Birkhäuser,Springer, 2012. [14] T. Ernst, q-Leibniz functional matrices with applications to q-Pascal and q-Stirling matrices. Adv. Stud. Contemp. Math. 22 (2012), 537–555. [15] T. Ernst, q-Pascal and q-Wronskian matrices with implications to q-Appell polynomials. J. Discrete Math. (2013), Article ID450481, 1–10. [16] T. Ernst, q-Pascal and q-Bernoulli matrices and umbral approach. Department of Mat- hematics Uppsala Universty D.M. Report 2008:23 (2008). [17] T. Ernst, Factorizations for q-Pascal matrices of two variables. Spec. Matrices 3 (2015), 207–213. [18] K. Kaygısız and A. S¸ahin, A new method to compute the terms of generalized order-k Fibonacci numbers. J. Number Theory 133 (2013), 3119–3126. [19] D.S. Kim and T.K. Kim, q-Bernoulli polynomials and q-umbral calculus. Sci. China Math. 57 (2014), 1867–1874. [20] D.S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus. J. Inequal. Appl. 2014: 1, (2014). [21] G.Y. Lee and J.S. Kim, The linear algebra of the k-Fibonacci matrix. Linear Algebra Appl. 373 (2003), 75–87. [22] G.Y. Lee, J.S. Kim and S.H. Cho, Some combinatorial identities via Fibonacci numbers. Discrete Appl. Math. 130 (2003), 527–534. [23] G.Y. Lee, J.S. Kim and S.G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. Fibonacci Quart. 40 (2002), 203–211. [24] T. Mansour, J.L. Ram´ırez and M. Shattuck, A generalization of the r-Whitney numbers of the second kind. J. Comb. 8(2017), 1, 29–55. [25] I. Mez˝oand J.L. Ram´ırez, The linear algebra of the r-Whitney matrices. Integral Trans- forms Spec. Funct. 26(2015), 3, 213–225. [26] A.R. Moghaddamfar, S. Navid Salehy and S. Nima Salehy, A matrix-theoretic perspective on some identities involving well-known sequences. Bull. Malays. Math. Sci. Soc. (2015), 1–14. [27] J.L. Ram´ırezand M. Shattuck, Generalized r-Whitney numbers of the first kind. Ann. Math. Inform. 46 (2016), 175–193. [28] J.L. Ram´ırezand M. Shattuck, (p, q)-Analogue of the r-Whitney-Lah numbers. J. Integer Seq. 19 (2016), Article 16.5.6. [29] J.L. Ram´ırez and V. Sirvent, A q-analogue of the biperiodic Fibonacci sequence. J. Integer Seq. 19 (2016), Article 16.4.6. [30] A. S¸ahin, On the q analogue of Fibonacci and Lucas matrices and Fibonacci polynomials. Util. Math. 100 (2016), 113–125. [31] S. Stanimirović,P. Stanimirović,M. Miladinovićand A. Ilić, Catalan matrix and related combinatorial identities. Appl. Math. Comput. 215 (2009), 796–805. [32] S. Stanimirović,J. Nikolov and I. Stanimirović, A generalization of Fibonacci and Lucas matrices. Discrete Appl. Math. 156 (2008), 2606–2619. [33] N. Tuglu and S. Ku¸s, q-Bernoulli matrices and their some properties. Gazi Univ. J. of Science 28 (2015), 269–273. [34] S.-L. Yang and Z.-K Liu, Explicit inverse of the Pascal matrix plus one. Int. J. Math. Math. Sci. Article ID 90901 (2006), 1–7. [35] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties. Discrete Appl. Math. 154 (2006), 1622–1632. 19 Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices 417

[36] Z. Zhang, The linear algebra of the generalized Pascal matrix. Linear Algebra Appl. 250 (1997), 51–60. [37] Z. Zhang and X. Wang, A factorization of the symmetric Pascal matrix involving the Fibonacci matrix. Discrete Appl. Math. 155 (2007), 2371–2376. [38] D. Zheng, q-analogue of the Pascal matrix. Ars Combin. 87 (2008), 321–336.

Received 10 January 2016 Universidad Sergio Arboleda, Departamento de Matemáticas, Bogotá,Colombia [email protected] Universidad Nacional de Colombia, Departamento de Matemáticas, Bogotá,Colombia [email protected] Gaziosmanpa¸saUniversity, Faculty of Education, Tokat, Turkey [email protected]; [email protected]