SOME RESULTS ON -ANALOGUE OF THE BERNOULLI, EULER AND FIBONACCI MATRICES GERALDINE M. INFANTE, JOSÉ L. RAMÍREZ and ADEM ŞAHİN Communicated by Alexandru Zaharescu In this article, we study -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 200 Subect Classification: 5A24, B68, 5A09, 5A5. Key words: -Bernoulli matrix, -Euler matrix, -Fibonacci matrix, -analogues, combinatorial identities.. INTRODUCTION Matrix theory is recently used in the study of several combinatorial seuences. 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 seuences such as Fibonacci and Lucas seuence [22,23,32,37, -Fibonacci numbers [2, Catalan numbers [3, Stirling numbers and their generalizations [8,,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 -Bernoulli numbers [, 9, - Euler numbers [20 and -Fibonacci numbers [2; for more details about these seuences see [3. The -Bernoulli polynomials, B n, (x), are defined by the exponential generating function F (x, t, ) : te (xt) e (t) B n, (x) tn [n! n0 ( t < 2π), where the -exponential function e (x) is defined by x n e (x) : [n! (0 < <, x < ), (( )x; ) n0 MATH. REPORTS 9(69), 4 (207), 399 47
400 Geraldine M. Infante, José L. Ramírez and Adem Şahin 2 with (a; ) : ( a ), [n ++ + n and [n! [ [2 [n. 0 In particular, if x 0 in B n, (x), we obtain the -Bernoulli numbers B n, (cf. [3). The -exponential function satisfies the addition formula e (x + y) e (x)e (y) provided that yx xy (-commutative variables). For a real or complex parameter α, the generalized -Bernoulli polynomials B n, (α) (x) are defined by the generating function () ( G(x, t, α, ) : t e (t) ) α e (xt) n0 B (α) n, (x) tn [n! ( t < 2π, α C). Clearly, B n,(x) () B n, (x) and B n,(0) () B n, are the -analogue of the Bernoulli polynomials and numbers, respectively. The -binomial coefficient is defined as n (; ) n :, (; ) (; ) n and n (a; ) n : ( a ). 0 Another way to write the -binomial coefficient is n [n! [![n!. From () we can derive the following euation n (2) B n, (α+β) (x + y) B (α), (x)b(β) n, (y) i0 0 provided that x commutes with y, i.e., if yx xy. Indeed, ( ) B (α+β) i, (x + y) ti [i i0! t α+β e ((x + y)t) e (t) ( ) t α ( ) t β e (xt) e (yt) e (t) e (t B (α) ti i, (x) B (β) ti i, (y) [i! [i! i0
3 Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 40 ( i i0 l0 ) i B (α) l l, (x)b(β) i l, (y) t i [i!. The result follows by comparing the coefficients. In particular, upon setting β 0 in identity (2) and interchanging x and y, we obtain Specifically, if α B (α) n, (x + y) (3) B n, (x + y) and if y 0 B n, (x) 0 0 0 n B (α), (y)xn. n B, (y)x n, n B n, x. The first few -Bernoulli polynomials are B 0, (x), B, (x) x +, 2 B 2,(x) x 2 x+ ( + ) ( 2 + + ), B 3, (x) x 3 2 + + x 2 + 2 + + x ( ) 3 ( + ) 2 ( 2 + ), B 4, (x) x 4 ( 2 + ) ( x 3 + 2 2 + ) x 2 + 3 4 + + x 0 8 2 7 6 + 4 + ( + ) 2 ( 2 + + ) ( 4 + 3 + 2 + + ). The -Euler polynomials E n, (x) are defined by means of the following generating function (cf. [20) E n, (x) tn [n! : 2 e (t) + e (xt) ( t < π). n0 In the special case, x 0, E n, (0) : E n, are called the -Euler numbers. The first few -Euler numbers are E 0,, E, 2, ï? 2 E 2, 4, E 3, 8 (+) ( 2 3 + ), E 4, 6 ( )( + ) ( 2 4 + ) ( 2 + + ),. On the other hand, there exist several slightly different -analogues of the Fibonacci seuence, among other references, see, [2 4, 9, 2, 29. In particular,
402 Geraldine M. Infante, José L. Ramírez and Adem Şahin 4 we are interested in the polynomials introduced by Cigler [2. The -Fibonacci polynomials, F n, (x, s), are defined as follows F n, (x, s) : n 2 0 n (+ 2 ) x n 2 s. In particular, if we tae x s, we obtain the -Fibonacci numbers F n,. The first few -Fibonacci number are F 0, 0, F,, F 2,, F 3, +, F 4, 2 + +, F 5, 2 3 + 2 + +, F 6, 5 + 2 4 + 2 3 + 2 + +, F 7, 7 + 2 6 + 3 5 + 2 4 + 2 3 + 2 + +,... It is clear that if we recover the well-nown Fibonacci seuence. In the present article, we study three families of matrices, the -Bernoulli matrix, the -Euler matrix and the -Fibonacci matrix. Then we obtain several results that generalize the classical case ( ). The outline of this paper is as follows. In Section 2, we introduce the generalize -Bernoulli matrix, then we derive some basic identities, in particular we find its inverse matrix. In Section 3, we find a factorization of the -Bernoulli matrix in terms of the -Pascal matrix. In Section 4, we show a relation between the -Pascal matrix plus one and the -Euler matrix. Finally, in Section 5 we introduce the -Fibonacci matrix and we find its inverse matrix and some interesting factorizations. In particular, we show a relation between the -Bernoulli polynomial matrix and the -Fibonacci matrix. 2. THE -ANALOGUE OF THE GENERALIZED BERNOULLI MATRIX We define the -analogue of the generalized (n + ) (n + ) Bernoulli matrix B n, (α) (x) : B (α) (x) : [B (α) (x)(0 i, n), where B (α) i, (x) i, {[ i B(α) i, (x), if i ; The matrices B n,(x) () : B () (x) : B (x) and B n,(0) () : B (0) : B are called the -Bernoulli polynomial matrix and the -Bernoulli matrix, respectively. For more information about this matrix see [3, 6. Note that this matrix is different from one recently studied by Tuglu and Kuş, [33. In particular, if we recover the generalized Bernoulli matrix [35.
5 Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 403 Example. For n 3, we have B 3, (x) 0 0 0 x + 0 0 x 2 x + 2 ( + )x 0 3 +2 2 +2+ x 3 (2 ++) + x 2 + 2 ( )3 + x (+) 2 ( 2 +) (2 ++)(x x 2 )+ 2 + (2 ++)x 2 ++ + and 0 0 0 + 0 0 B 3, 2 0 3 +2 2 +2+. ( )3 2 + 2 ++ + (+) 2 ( 2 +) Theorem 2. If x commutes with y, then the following eualities hold for any α and β (4) [ i B (α+β) (x + y) B (α) (x)b (β) (y) B (β) (y)b (α) (x). We need the following -binomial identity. Lemma 3. The following identity holds for any positive integers i,, i i i Proof. From the definition of -binomial coefficient we get [i! [i! [![i! [![i +! [i![! [![![i![! [i! [![i!. [! [![! Proof of Theorem 2. From above lemma and Identity (2) we have (B (α) (x)b (β) (y)) i, i i i B (α) i, (x) B (β), (y) i [ i [ i B (α) i i, (x)b(β), (y) Then Euation (4) follows. 0. B (α) i, (x)b(β), (y) B (α+β) i, (x + y) (B(α+β) (x + y)) i,.
404 Geraldine M. Infante, José L. Ramírez and Adem Şahin 6 If we obtain Theorem 2. of [35. Corollary 4. If the variables commute then the following euality holds for any integer 2 (5) B (α +α 2 + +α ) (x + x 2 + + x ) B (α ) (x )B (α 2) (x 2 ) B (α ) (x ). Proof. We proceed by induction on. From Theorem 2 the euality clearly holds for 2. Now suppose that the result is true for all i <. We prove it for. B (α +α 2 + +α ) (x + x 2 + + (x + x )) By Theorem 2, we get B (α ) (x )B (α 2) (x 2 ) B (α +α ) (x + x ) B (α ) (x )B (α 2) (x 2 ) B (α +α ) (x + x ). B (α ) (x )B (α 2) (x 2 ) B (α ) (x )B (α) (x ). If we tae x x 2 x x and α α 2 α α, then (B (α) (x)) B (α) (x). In particular, if α, then (B (x)) B () (x); if x 0, (B (α) ) B (α). Finally, if α, x 0 we obtain B B (), i.e., a formula of the powers of the -Bernoulli matrix. Now, we study the inverse matrix of the -Bernoulli matrix. Let D [d i, (0 i, n) be the (n + ) (n + ) matrix defined by { [ i [i + d i,, if i ; We need the following lemma [20. Lemma 5. For any positive integer n n B n, [n!δ n,0, [ + 0 where δ n,m is the Kronecer delta symbol. Theorem 6. The inverse matrix of the -Bernoulli matrix B is the matrix D, i.e, B D. Furthermore, (B () ) D. Proof. From above lemma, we get (B D ) i, i B i, [ + [
7 Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 405 B i 0 [ + i [ + B i, [ i B i, [n!δ i,0. This is only when i, and 0 otherwise. Therefore B D I, i.e., D. Moreover, if α and x 0 we get (B () ) (B ) (B ) D. If we obtain Theorem 2.4 of [35. 3. THE -ANALOGUE OF THE GENERALIZED BERNOULLI MATRIX AND THE GENERALIZED -PASCAL MATRIX In this section, we show some relations between the -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 inds of generalizations of the Pascal matrix have been presented in the literature. In particular, we are interested in its -analogue; see, e.g., [3 7, 38. The (n + ) (n + ) -Pascal matrix P n : P n, : [p i, (0 i, n) is defined with the -binomial coefficients as follows {[ i p i, :, if i ; For any nonnegative integers n and, the (n + ) (n + ) matrices I n, S n (), D n () and P n () are defined by (I n ) i, : diag(,,..., ), { (S n () (i ), if i ; ) i, :, if i ; (D n () ) i,, if i ; (P () n ) i, : (P n ) i, (i ). (0 i, n),
406 Geraldine M. Infante, José L. Ramírez and Adem Şahin 8 Clearly, P n (0) P n. Furthermore we need the (n + ) (n + ) matrices P () 0 n : 0 P () n In 0 F : 0 D (n ), ( n ) and F n : D n (0), In 0 G : 0 S (n ), ( n ) and G n : S n (0). It is not difficult to see that (D n () ) S n () and F G. Moreover, for any nonnegative integer, we have the following factorization [38 D n () P n () P (+) n. By the above euation and the definition of the matrix F, we get F F 2 F n P n I n or P n Fn Fn F. Therefore, the -Pascal matrix P n can be factorized as follows P n G n G n G. Moreover, it is clear that the inverse of the -Pascal matrix can be factorized as Pn F F 2 F n P n, where (P n ) i, : ( ) i (P n ) i, (i 2 ). The (n+) (n+) generalized -Pascal matrix P n [x : [p i, (0 i, n), is defined by [38 p i, {[ i xi, if i ; Example 7. For n 4, the generalized -Pascal matrix is 0 0 0 0 x 0 0 0 P 4 [x x 2 ( + )x 0 0 x 3 ( 2 + + )x 2 ( 2 + + )x 0 x 4 ( + )( + 2 )x 3 ( + 2 )( + + 2 )x 2 ( + )( + 2 )x In [7, Ernst studied a generalization of this matrix. Theorem 8. If x commutes with y, then we have the following identity In particular, B (x) P n [xb. B (x + y) P n [xb (y) P n [yb (x).
9 Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 407 Proof. From Euation (3), we have (P n [xb (y)) i, i i [ i x i [ B, (y)x i B, (y) [ i i 0 Similarly, we have B (x + y) P n [yb (x). In particular, if we obtain Theorem 3. of [35. B, (y)x i B i, (x + y) (B (x + y)) i,. Example 9. If n 2 we obtain the following factorization 0 0 B 2, (x) x + 0 x 2 x + 2 ( + )x 3 +2 2 +2+ 0 0 0 0 x 0 + 0 P 2 [xb 2,. x 2 ( + )x 2 3 +2 2 +2+ Let S () n [x and D () [x be the (n + ) (n + ) matrices defined by n (S n () [x) i, (S () n ) i, x i and (D n () [x) i, (D () n ) i, x i. For n, we define the following matrices In 0 F [x : 0 D () n [x, ( n ) and F n [x : D n (0) [x, In 0 G [x : 0 S () n [x, ( n ) and G n [x : S n (0) [x. Clearly, F [x G [x, n. We need the (n + ) (n + ) matrix By definition, we have [38 I n [x : diag(, x,..., x n ). S () n [x I n [xs () n In [x, G [x I n [xg In [x, P n [x I n [xp n In [x.
408 Geraldine M. Infante, José L. Ramírez and Adem Şahin 0 Moreover, the generalized -Pascal matrix P n [x can be factorized by the matrices G [x as follows Moreover, P n [x I n [xg n G n G In [x (I n [xg n In [x)(i n [xg n In [x) (I n [xg In [x) G n [xg n [x G [x. Pn [x F n [xf n [x F [x P n [x, where (P n [x) i, : (P n ) i, (i 2 ) ( x) i. From above factorizations we have the following theorem. Theorem 0. For any x we have the following identities B (x) G n [xg n [x G [xb, B (x) D F n [xf n [x F [x. If we obtain Corollary 3.3 of [35. 4. INVERSE MATRIX OF THE -PASCAL MATRIX PLUS ONE In this section, we give an explicit formula to the inverse matrix of the -Pascal matrix plus one, and we show the relation between this inverse matrix and the -Euler numbers. Yang and Liu [34 studied the case. Lemma. For any n 0, we have n E, (x) + 2 2 E n,(x) x n, 0 where E n, (x) are the -Euler polynomials defined by E n, (x) tn [n! 2 e (t) + e (xt), ( t < π). n0 n0 Proof. From definition of -Euler numbers we get ( ) E n, (x) tn 2 (e (t) + ) [n! e (t) + e (xt)(e (t) + ) 2e (xt). Therefore, ( ) n E i, (x) + E n, (x) i n0 i0 t n [n! By comparing coefficients, we get the desired result. n0 tn 2x n [n!.
Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 409 Lemma 2. If x commutes with y, then for any n 0, we have n E n, (x + y) E i, (x)y n i. i In particular, if x 0 we have E n, (y) i0 i0 n E i, y n i. i Proof. From definition of -Euler numbers and since x, y are -commutative variables, we get n0 E n, (x + y) tn [n! 2 e (t) + e ((x + y)t) ( ) E n, (x) tn e (yt) [n! n0 n0 2 e (t) + e (xt)e (yt) ( ) n E i, (x)y n i i i0 By comparing coefficients, we get the desired result. t n [n!. We define the -analogue of the (n + ) (n + ) Euler matrix E n, : [E i, (0 i, n), where E i, {[ i E i,, if i ; Theorem 3. For all positive integer n the following identity holds (P n + I n ) 2 E n,. Proof. From Lemma we have the following matrix euation 2 (P n + I n )E n, (x) X n, where E n, (x) and X n are n matrices defined by E n, (x) [E 0, (x), E, (x),..., E n, (x) T, X n [, x,..., x n T. From Lemma 2 we have the following matrix euation E n, (x) E n, X n. Therefore 2 (P n + I n )E n, I n. If we obtain the main result in [34.
40 Geraldine M. Infante, José L. Ramírez and Adem Şahin 2 5. SOME RELATIONS BETWEEN -PASCAL MATRIX AND -FIBONACCI MATRIX In this section, we give some relations between -Pascal matrix and - Fibonacci matrix. Moreover, we obtain the inverse matrix of the -Fibonacci matrix. We define the -analogue of the (n + ) (n + ) Fibonacci matrix F n, : F : [f i, (0 i, n), where { F i,, if i ; f i, A generalization of the -Fibonacci matrix and its inverse were recently studied in [30. Example 4. The -Fibonacci matrix for n 4 is 0 0 0 0 0 0 0 F 4, + 0 0 2 + + + 0. 2 3 + 2 + + 2 + + + We need the following auxiliary seuence and matrix. seuence defined by N 0, N and n N n N n + ( ) n F n +2, N, n 2. The first few terms of N n are Let N n be the,,, 2, 2 2 2 3, 5 +2 4 4 3 + 2, 7 2 6 5 +7 4 3 3,.... The (n + ) (n + ) lower Hessenberg matrix C n : [c i, (0 i, n) is defined by { F i +2,, i + 2 ; c i, Lemma 5 ([6). Let A n be an n n lower Hessenberg matrix for all n and define det(a 0 ). Then, det(a ) a and for n 2 n n det(a n ) a n,n det(a n ) + [( ) n r a n,r ( a,+ ) det(a r ). r Lemma 6. For n, det(c n ) N n. r
3 Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 4 Proof. We proceed by induction on n. The result clearly holds for n. Now suppose that the result is true for all positive integers less than or eual to n. We prove it for n +. In fact, by using Lemma 5 we have n det(c n+ ) c n+,n+ det(c n ) + ( ) n+ i a n+,i a,+ det(c i ) det(c n ) + N n + i i [ ( ) n+ i F n i+3, det(c i ) i [ ( ) n+ i F n i+3, N i Nn+. i We need the following Theorem of Chen and Yu [0. Theorem 7. Let H [h i, be an n n lower Hessenberg matrix, 0 0 H. H 0 and H [αn [L [ n n h β T. Then n (n+) (n+) (6) det(h) ( ) n h det( H), (7) L H + h αβ T, (8) Hα + he n 0, where e n is nth column of matrix I n. Lemma 8 ([8). Let e be the first column of the identity matrix I n and L, β, h be the matrices described in the Theorem 7. Then, (9) β T H + he 0. Theorem 9. Let F n, be the (n+) (n+) lower triangular -Fibonacci matrix, then its inverse matrix is given by ( ) i N i, i > ; (F n, ) [b i,, i ;
42 Geraldine M. Infante, José L. Ramírez and Adem Şahin 4 Proof. It is obvious that, 0 0 (C n ). C n 0 F n+,. Let us construct the matrix (C n ) [αn [ [L n n h β T n obtain the entries. () By using (6) we obtain h ( )n det(c n) det( C n) ( ) n det(c n ), (2) We obtain matrices [α and [β by using (8) and (9): det C [α (C n ) ( ) n det(c n )e n. ( ) n 2 det C n 2 ( ) n det C n and [β T ( ) n det(c n )e (C n ). Therefore (n+) (n+) [β T [ ( ) n det C n ( ) n 2 det C n 2 det C. (3) We obtain matrix [L by using (7) and Lemma 6: [L (C n ) + (( ) n det C n ) αβ T 0 0. det C............. 0. ( ) n 2 det C n 2 det C Conseuently, combining these three steps and using Lemma 6, we obtain the reuired result. In particular, if we obtain the inverse matrix of the Fibonacci matrix [23. Lemma 6 and Theorem 9 were recently generalized in [30. Example 20. The inverse matrix of F 4, is 0 0 0 0 (F 4, ) 0 0 0 0 0 2 0. 2 2 2 3 2 and
5 Some results on -analogue of the Bernoulli, Euler and Fibonacci matrices 43 The (n + ) (n + ) lower triangular matrix R n, : [r i,, (0 i, n) is defined by { i 0 r i, ( ) i + N, if i ; Theorem 2. The following euality holds for any positive integer n P n, R n, F n,. Proof. Note that it suffices to prove that P n, (F n, ) R n,. For i 0 we have p i, b, 0 0 ( ) N i ( ) N i [ i ( ) + 0 N r i, and it is obvious that r i, 0 for i > 0, which implies that P n, (F n, ) R n,, as desired. In particular, if we obtain Theorem 2. of [37. Example 22. 0 0 0 0 0 0 0 P 4, + 0 0 2 + + 2 + + 0 3 + 2 + + 4 + 3 + 2 2 + + 3 + 2 + + 0 0 0 0 0 0 0 0 2 0 0 3 3 2 2 + 0 2 5 4 5 3 2 4 3 3 2 4 + 2 3 + 2 + 0 0 0 0 0 0 0 + 0 0 2 + + + 0 R 4, F 4,. 2 3 + 2 + + 2 + + +
44 Geraldine M. Infante, José L. Ramírez and Adem Şahin 6 The (n + ) (n + ) lower triangular matrix S n, : [s i,, (0 i, n) is defined by i 0 ( )+ + N i, if i, i odd; s i, i 0 ( ) + N i, if i, i even; Theorem 23. The following euality holds for any positive integer n P n, F n, S n,. Proof. The proof runs lie in Theorem 2. In particular, if we obtain the Theorem 2. of [22. Example 24. 0 0 0 0 0 0 0 P 4, + 0 0 2 + + 2 + + 0 3 + 2 + + 4 + 3 + 2 2 + + 3 + 2 + + 0 0 0 0 0 0 0 + 0 0 2 + + + 0 2 3 + 2 + + 2 + + + 0 0 0 0 0 0 0 0 0 0 2 + 2 + 2 0 4, S 4,. 2 2 3 2 2 + 3 + 2 + 3 + 4 + 2 + 3 Finally, we show a relation between the -Fibonacci matrix and the - Bernoulli matrix. The (n + ) (n + ) lower triangular matrix T n, : [t i,, 0 i, n is defined by t i, {[ i B i,(x) + i ( )i [ N i B, (x), if i 0 Theorem 25. The following euality holds for any positive integer n B n, (x) F n, T n,.
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