Cent Eur J Math 9(6 0 403-40 DOI: 0478/s533-0-0089-9 Central European Journal of Mathematics The -Fibonacci matrix and the Pascal matrix Research Article Sergio Falcon Department of Mathematics and Institute for Applied Microelectronics (IUMA, University of Las Palmas de Gran Canaria, 3507-Las Palmas de Gran Canaria, Spain Received March 0; accepted 8 August 0 Abstract: We define the -Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the -Fibonacci numbers Then we give two factorizations of the Pascal matrix involving the -Fibonacci matrix and two new matrices, L and R As a consequence we find some combinatorial formulas involving the -Fibonacci numbers MSC: 5A36, C0, B39 Keywords: Pascal matrix -Fibonacci numbers Factorization of a matrix Versita Sp z oo Introduction Many generalizations of the Fibonacci sequence have been introduced and studied [6, 9] Here we use the -Fibonacci numbers defined as follows [4, 5]: for any positive real number, the -Fibonacci sequence, say {F,n } n N, is defined recurrently by F,n+ = F,n + F,n, n, ( with the initial conditions F,0 = 0, F, = Following [, 6, 8, 9, ], we define the n n, -Fibonacci matrix as the unipotent lower triangular Toeplitz matrix F n ( = [f i, ] i,=,,n defined with entries f i, = F,i + if i, 0 otherwise That is, F, F,3 F, F n ( = F,4 F,3 F, F,n F,n F,n F,n 3 E-mail: sfalcon@dmaulpgces 403
The -Fibonacci matrix and the Pascal matrix From now on, we will designate the matrix F n ( as F n Lee et al [8] discussed the factorizations of the Fibonacci matrix corresponding to the classical Fibonacci sequence, and the eigenvalues of the symmetric Fibonacci matrix F n Fn T It is not difficult to prove that the inverse matrix of the -Fibonacci matrix introduced above, is given by the lower triangular matrix Fn = [f i,] i,=,,n where if = i, f i, if = i, = if = i, 0 otherwise That is, n ( = F 0 0 0 0 Let Fn matrix be the matrix obtained from F n by deleting the first row [0] The negative matrix of the matrix F n H(, n = 0, 0 0 0 and the sequence of principal minors is precisely the -Fibonacci sequence {F,n } n is [3] the On the other hand, the well-nown n n Pascal matrix P n = [p i, ] i,=,,n is defined by p i, = ( i if i, 0 otherwise That is, P n = 3 3 ( n ( n ( n ( n ( n 0 3 n The inverse of the Pascal matrix is P n = [p i,] i,=,,n with p i, = ( i+( i if i, 0 otherwise, see [] In this paper, following [7, ], we will show two factorizations of the Pascal matrix involving the -Fibonacci matrix Let us define the matrix L n ( = [l i, ] i,=,,n by For instance, We will designate the matrix L n ( as L n l i, = ( i ( i L 5 ( = 3 3 5 3 4 ( i 3 ( In this note, we use the -Fibonacci sequence to extend the results of [7, ] Note that if = in ( then l i, is the matrix defined in [] The recursive formula for matrix L n is that each entry is the sum of the elements to the left and above in the preceding row; that is, for, l i, = l i, + l i,, which is the same rule as used in the Pascal matrix 404
S Falcon First factorization of the Pascal matrix Theorem For every n N, the matrix P n may be written in the form P n = F n L n Proof It is enough to prove that Fn P n = L n Since the left-hand matrices are lower triangular, the product is a lower triangular matrix as well In addition, since the elements of the main diagonal of P n and Fn are, their product has also there That is, l i, = if = i, 0 if > i Moreover, for i, and by relation ( and the recurrence relation (, l i, = n h= f i,h p h, occurs and the proof is complete Corollary For every n 3 and n the following holds: n [( + r r=0 ( + r ( + r 3 ] F,n + r = ( n Proof Note that the entry p n, in matrix P n = F n L n is of the form n p n, = F,n + r l +r,, r=0 from which the relation follows In particular, for =, taing into account that the terms of the first column of matrix L n are of the form {,,,, }, p n, =, we have = F,n + F,n ( (F,n + F,n 3 + + F, and so we arrive at the next result Corollary 3 F,r = (F,n + F,n+ r=0 This corollary appears as [5, Proposition 8] Lemma 4 For n, = [( ( n n ( n ] F, = (3 Proof By a suitable change of indices, the proposed identity is equivalent to ( n F,+ = + [ ( ( ] n n + F,+ + + This equation is trivial for n = 0 and n = because LHS = RHS = + ( F, + 0 [ ( F, = +, ( ] F, = + ( + 405
The -Fibonacci matrix and the Pascal matrix Let us suppose n > Then LHS = ( n F, + 0 = + = ( n n = ( n F,+ + F, + = ( n n n F,n+ = + ( n F, = + n = [ ( n (F, + F, + F,n + F,n ( n + ( ] n + F,+ = RHS + Lemma 5 For n 3, [( ( ( ] ( n n n F, = (n + n 3 + (4 =3 Proof By induction Let us denote as a n, the coefficient of F, in (4 For n = 3, (LHS 3 = [( ( ( ] F,3 = F 3,3 = + = (RHS 3 and the relation holds Let us suppose the relation is true for any m n Since ( m ( i = m ( i + m i, then n+ (LHS n+ = a n+, F, = =3 = (RHS n + = (RHS n + n+ a n, F, + a n, F, =3 =3 a n, F,+ = (RHS n + = ( (n ( = ( + (RHS n + (n ( a n, F, + F, = (n (n 3 (n (n 3 F, + (RHS n + a n, F, = + = (n ( + n + = (RHS n+ having used the formula (3 Theorem 6 For every n N, L n = [l i,] i,=,,n with l i, = i r= ( i ( i+r F,r + r Proof It is enough to use the fact that P n F n = L n 3 Second factorization of the Pascal matrix Now, following [7, ], we will show another factorization of the Pascal matrix involving the -Fibonacci matrix Let us define the matrix R n ( = [r i, ] i,=,,n by r i, = ( i ( i ( i + 406
S Falcon For instance, R 6 ( = 3 3 3 5 4 6 5 4 4 9 5 5 0 5 0 9 5 5 From now on, we will write R n for the matrix R n ( The following theorem holds: Theorem 3 For every n N, P n = R n F n Proof It is enough prove that P n F n = R n Corollary 3 For every n, we have r= [( n r ( n r ( ] n F,r + = r + ( n In particular, for =, we have the following corollary which coincides with Lemma 4 Corollary 33 r= [( n r ( n r ( ] n F, = r + Theorem 34 The inverse matrix of R n is R n = [r i,] i,=,,,n with entries r i, = i ( i + r ( i+r F,r + r= Proof It is enough to use the fact that F n P n = R n 4 Product of the -Fibonacci matrix and its transpose In this section, we will study the matrix obtained by means of the product of the -Fibonacci matrix and its transpose, its relation with the preceding matrices F n, P n, L n, and R n, and some of its properties Let S n = [s r,l ] r,l=,,n be the product matrix S n = F n F T n Then, the entries of this matrix S n are s r,l = c F,r F,l, where c = min (r, l Taing into account that the product of a matrix and its transpose is a symmetric matrix, we have s r,l = s l,r, and we can suppose r l Consequently, r, and so s r,l = l F,r F,l (5 407
The -Fibonacci matrix and the Pascal matrix This formula also can be expressed as s r,l = l F, F,r l+ (6 The entries of the first line (row or column of this matrix are the -Fibonacci numbers, s r, = s,r = F,r, while the entries of the second line are s r, = s,r = F,r+ Moreover, taing into account that P n = R n F n, we can deduce S n = R n P n Fn T or S n = P n L n Fn T Theorem 4 Elements s r,l of the matrix S n fulfil the relations s,0 = s 0, = 0, (7 s r,r = F,rF,r+, (8 s r,l = s r,l + s r,l, r < l, (9 s r,l = s r,l + s r,l, r > l (0 Proof Equation (7: For r = l, the entries of main diagonal of S n are obtained and s r,r = r = F,r = F,rF,r+ by applying a formula found in [] for the sum of the squares of -Fibonacci numbers Equation (9: Let l = r + h Then, s r,l = s r,r+h = = r F,r F,r+h = r F,r F,r+h + r ( F,r F,r+h + F,r+h r F,r F,r+h = s r,r+h + s r,r+h = s r,l + s r,l Finally, taing into account that matrix S n is symmetric the formula (0 can be obtained Given that F n =, we have S n = For n = 0 and =, the matrix Q 0 defined in [8] is found In the sequel, we will indicate the general expression of terms of the matrix S n We have seen the first two lines of S n are -Fibonacci numbers, the entries of the main diagonal are s r,l = F,rF,r+ and s r,l = s l,r In the following theorem, we will give a very simple expression for these last entries of the matrix S n, which have the form m l= F,lF,r (m l Theorem 4 For h 0, we have F, F,+h = ( F,n+ F,n+h ( n + F,h ( Proof We will prove this theorem by induction For n =, in this formula is (LHS = F, F,+h = F, F,+h = F,+h, (RHS = F,F,+h = F,+h = F,+h Let us suppose this formula is true until n, that is n F, F,+h = ( F,n F,n +h ( n + F,h 408
S Falcon Then n F, F,+h = F, F,+h + F,n F,n+h = ( F,n F,n +h ( n + F,h + F,n F,n+h = ( F,n F,n++h ( n + F,h = ( F,n F,n++h + ( n F,h ( n + F,h = ( F,n+h F,n+ ( n + F,h after applying to the penultimate expression the D Ocagne identity [4, 5]: F,m F,n+ F,m+ F,n = ( n F,m n Theorem 43 (main result For r l, s r,l = F,rF,l+ ( F,r F,l+ F,r l if l is odd, if l is even ( Proof If in ( we change n by l and h by r l, we obtain the entries s r,l of the matrix S n, according to (6: s r,l = l F, F,+r l = ( F,r F,l+ ( l + F,r l Then ( follows Taing into account formulas (5 and (, the following two corollaries can be deduced Corollary 44 If r l, then l F,r F,l = F,rF,l+ ( F,r F,l+ F,r l if l is odd, if l is even These formulas have been proved in [] Corollary 45 The inverse matrix of S n is the product (Fn T Fn, so, S n = + 0 0 0 0 0 + 0 0 0 0 0 + 0 0 0 0 0 0 + 0 0 0 0 0 + 0 0 0 For =, the matrix Q 0 already defined in [8] is found The decomposition of the defined positive symmetric matrix S by means of the product of a lower triangular matrix F n and its transpose matrix is a Cholesy decomposition, so it is unique 409
The -Fibonacci matrix and the Pascal matrix Acnowledgements The author sincerely thans the anonymous referee for valuable comments and suggestions, which significantly improved the quality of this paper This wor has been supported in part by CICYT Proect number MTM008-05866-C03-0/MTM from Ministerio de Educación y Ciencia of Spain References [] Call GS, Velleman DJ, Pascal s matrices, Amer Math Monthly, 993, 00(4, 37 376 [] Falcón S, On sequences of products of two -Fibonacci numbers, Int J Contemp Math Sci (in press [3] Falcón S, Some tridiagonal matrices and the -Fibonacci numbers Appl Math Comput (in press [4] Falcón S, Plaza Á, On the Fibonacci -numbers, Chaos Solitons Fractals, 007, 3(5, 65 64 [5] Falcón S, Plaza Á, The -Fibonacci sequence and the Pascal -triangle, Chaos Solitons Fractals, 007, 33(, 38 49 [6] Lee G-Y, Kim J-S, The linear algebra of the -Fibonacci matrix, Linear Algebra Appl, 003, 373, 75 87 [7] Lee G-Y, Kim J-S, Cho S-H, Some combinatorial identities via Fibonacci numbers, Discrete Appl Math, 003, 30(3, 57 534 [8] Lee G-Y, Kim J-S, Lee S-G, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart, 00, 40(3, 03 [9] Lee G-Y, Lee S-G, Shin H-G, On the -generalized Fibonacci matrix Q, Linear Algebra Appl, 997, 5, 73 88 [0] Peart P, Woodson L, Triple factorization of some Riordan matrices, Fibonacci Quart, 993, 3(, 8 [] Zhang Z, The linear algebra of the generalized Pascal matrix, Linear Algebra Appl, 997, 50, 5 60 [] Zhang Z, Wang X, A factorization of the symmetric Pascal matrix involving the Fibonacci matrix, Discrete Appl Math, 007, 55(7, 37 376 40