Determinant and Permanent of Hessenberg Matrix and Fibonacci Type Numbers

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Gen. Math. Notes, Vol. 9, No. 2, April 2012, pp.32-41 ISSN 2219-7184; Copyright c ICSRS Publication, 2012 www.i-csrs.org Available free online at http://www.geman.

1 Gen. Math. Notes, Vol. 9, No. 2, April 2012, pp ISSN ; Copyright c ICSRS Publication, Available free online at Determinant and Permanent of Hessenberg Matrix and Fibonacci Type Numbers Kenan Kaygısız 1 and Adem Şahin 2 1 Department of Mathematics Faculty of Arts and Sciences Gaziosmanpaşa University 60240, Tokat, Turkey kenan.kaygisiz@gop.edu.tr 2 Department of Mathematics Faculty of Arts and Sciences Gaziosmanpaşa University 60240, Tokat, Turkey adem.sahin@gop.edu.tr (Received: / Accepted: ) Abstract In this paper, we obtain determinants and permanents of some Hessenberg matrices that give the terms of k sequences of generalized order-k Fibonacci numbers for k = 2. These results are important, since k sequences of generalized order-k Fibonacci numbers for k = 2 are general form of ordinary Fibonacci sequence, Pell sequence and Jacobsthal sequence. Keywords: Fibonacci Numbers, Jacobsthal Numbers, k sequences of generalized order-k Fibonacci numbers, Pell Numbers, Hessenberg Matrix. 1 Introduction Fibonacci numbers F n, Pell numbers P n and Jacobsthal numbers J n are defined by F n = F n 1 + F n 2 for n > 2 and F 1 = F 2 = 1, P n = 2P n 1 + P n 2 for n > 1 and P 0 = 0, P 1 = 1, J n = J n 1 + 2J n 2 for n > 2 and J 1 = J 2 = 1, respectively. Generalizations of these sequences have been studied by many researchers.

2 Determinant and Permanent of Hessenberg Matrix 33 Er [3] defined k sequences of generalized order-k Fibonacci numbers (ksokf) as; for n > 0, 1 i k f i k,n = k j=1 with boundary conditions for 1 k n 0, f i k,n = c j f i k,n j (1) { 1, if i = 1 n, 0, otherwise, where c j (1 j k) are constant coefficients, f i k,n is the n-th term of i-th sequence of order k generalization. and Example 1.1 f 1 k,n and f 2 k,n sequences are respectively. 0, 1, c 1, c 2 + c 2 1, 2c 1 c 2 + c 3 1, c c c 2 1c 2, c c 1 c c 3 1c 2, c c c 4 1c 2 + 6c 2 1c 2 2, c c 1 c c 5 1c c 3 1c 2 2, c c c 6 1c c 2 1c c 4 1c 2 2,... 1, 0, c 2, c 1 c 2, c c 2 1c 2, 2c 1 c c 3 1c 2, c c 4 1c 2 + 3c 2 1c 2 2, 3c 1 c c 5 1c 2 + 4c 3 1c 2 2, c c 6 1c 2 + 6c 2 1c c 4 1c 2 2, 4c 1 c c 7 1c c 3 1c c 5 1c 2 2,... A direct consequence of (1) is f 2 k,n = c 2 f 1 k,n 1, for n 0. (2) Remark 1.2 Let f i k,n, F n, P n and J n be ksokf (1), Fibonacci sequence, Pell sequence and Jacobsthal sequence, respectively. Then, (i) Substituting c 1 = c 2 = 1 for k = 2 in (1), we obtain f 1 k,n 1 = F n. (ii) Substituting c 1 = 2 and c 2 = 1 for k = 2 in (1), we obtain f 1 k,n 1 = P n. (iii) Substituting c 1 = 1 and c 2 = 2 for k = 2 in (1), we obtain f 1 k,n 1 = J n.

3 34 Kenan Kaygısız et al. Remark 1.2 shows that fk,n 1 is a general form of Fibonacci sequence, Pell sequence and Jacobsthal sequence. Therefore, any result obtained from fk,n 1 holds for other sequences mentioned above. Many researchers studied on determinantal and permanental representations of k sequences of generalized order-k Fibonacci and Lucas numbers. For example, Minc [7] defined an n n (0,1)-matrix F (n, k), and showed that the permanents of F (n, k) is equal to the generalized order-k Fibonacci numbers. The author of [5, 6] defined two (0, 1)-matrices and showed that the permanents of these matrices are the generalized Fibonacci and Lucas numbers. Öcal et al. [8] gave some determinantal and permanental representations of k-generalized Fibonacci and Lucas numbers, and obtained Binet s formula for these sequences. Yılmaz and Bozkurt [9] derived some relationships between Pell and Perrin sequences, and permanents and determinants of a type of Hessenberg matrices. In this paper, we give some determinantal and permanental representations of k sequences of generalized order-k Fibonacci numbers for k = 2 by using various Hessenberg matrices. These results are general form of determinantal and permanental representations of ordinary Fibonacci numbers, Pell numbers and Jacobsthal numbers. 2 The Determinantal Representations An n n matrix A n = (a ij ) is called lower Hessenberg matrix if a ij = 0 when j i > 1, i.e., A n = a 11 a a 21 a 22 a 23 0 a 31 a 32 a a n 1,1 a n 1,2 a n 1,3 a n 1,n a n,1 a n,2 a n,3 a n,n. (3) Theorem 2.1 [2] Let A n be an n n lower Hessenberg matrix for all n 1 and det(a 0 ) = 1. Then, and for n 2 det(a n ) = a n,n det(a n 1 ) + det(a 1 ) = a 11 n 1 r=1 n 1 ( 1) n r a n,r a j,j+1 det(a r 1 ). (4) j=r

4 Determinant and Permanent of Hessenberg Matrix 35 Theorem 2.2 Let f 1 2,n be the first sequence of 2SO2F and Q n = (q rs ) n n be a Hessenberg matrix defined by q rs = i r s. c r s+1 c (r s) 2, if 1 r s < 2, 0, otherwise, that is Then, Q n = c 1 ic i c 1 ic i c 1 ic c 1 ic i c 1. (5) det(q n ) = f 1 2,n, (6) where c 0 = 1 and i = 1. Proof. To prove (6), we use the mathematical induction on m. The result is true for m = 1 by hypothesis. Assume that it is true for all positive integers less than or equal to m, namely det(q m ) = f2,m. 1 Then, we have m det(q m+1 ) = q m+1,m+1 det(q m ) + ( 1) m+1 r m q m+1,r q j,j+1 det(q r 1 ) r=1 j=r m 1 = c 1 det(q m ) + ( 1) m+1 r m q m+1,r q j,j+1 det(q k,r 1 ) r=1 j=r + [( 1)q m+1,m q m,m+1 det(q k,m 1 )] = c 1 det(q m ) + [( 1)ic 2 i det(q k,m 1 )] = c 1 det(q m ) + c 2 det(q k,m 1 ) by using Theorem 2.1. From the hypothesis of induction and the definition of 2SO2F, we obtain det(q m+1 ) = c 1 f 1 2,m + c 2 f 1 2,m 1 = f 1 2,m+1. Therefore, (6) holds for all positive integers n.

5 36 Kenan Kaygısız et al. Example 2.3 We obtain f 1 2,6, by using Theorem 2.2 det(q 6 ) = det c 1 ic i c 1 ic i c 1 ic i c 1 ic i c 1 ic i c 1 = c c c 4 1c 2 + 6c 2 1c 2 2 = f 1 2,6. Theorem 2.4 Let f2,n 1 be the first sequence of 2SO2F and B n = (b ij ) n n be a Hessenberg matrix, where b ij = c 2, if j = i + 1,, if 0 i j < 2, c i j+1 c (i j) 2 0, otherwise, that is B n = c 1 c c 1 c c c 1 c c 1 Then, det(b n ) = f 1 2,n, where c 0 = 1. Proof. Since the proof is similar to the proof of Theorem 2.2 by using Theorem 2.1, we omit the detail. Example 2.5 We obtain f 1 2,4 by using Theorem 2.4 that det(b 5 ) = det = c c c 2 1c 2 = f 1 2,4. c 1 c c 1 c c 1 c c 1

6 Determinant and Permanent of Hessenberg Matrix 37 Corollary 2.6 If we rewrite Theorem 2.2 and Theorem 2.4 for c i = 1, then we obtain det(q n ) = F n+1 and det(b n ) = F n+1, respectively, where F n s are the ordinary Fibonacci numbers. Corollary 2.7 If we rewrite Theorem 2.2 and Theorem 2.4 for c 1 = 2 and c 2 = 1, then we obtain det(q n ) = P n+1 and det(b n ) = P n+1, respectively, where P n s are the Pell numbers. Corollary 2.8 If we rewrite Theorem 2.2 and Theorem 2.4 for c 1 = 1 and c 2 = 2, then we obtain det(q n ) = J n+1 and det(b n ) = J n+1, respectively, where J n s are the Jacobsthal numbers. 3 The Permanent Representations Let A = (a i,j ) be an n n matrix over a ring. Then, the permanent of A is defined by per(a) = n a i,σ(i), σ S n i=1 where S n denotes the symmetric group on n letters. Theorem 3.1 [8] Let A n be n n lower Hessenberg matrix for all n 1 and per(a 0 ) = 1. Then, per(a 1 ) = a 11 and for n 2 n 1 n 1 per(a n ) = a n,n per(a n 1 ) + a n,r a j,j+1 per(a r 1 ). (7) r=1 Theorem 3.2 Let f2,n 1 be the first sequence of 2SO2F and H n = (h rs ) be an n n Hessenberg matrix, where that is Then, h rs = H n = where c 0 = 1 and i = 1. i (r s). c r s+1 c (r s) 2 j=r, if 1 r s < 2, 0, otherwise, c 1 ic i c 1 ic i c c 1 ic i c 1 per(h n ) = f 1 2,n,. (8)

7 38 Kenan Kaygısız et al. Proof. This is similar to the proof of Theorem 2.2 using Theorem 3.1. Example 3.3 We obtain f 1 2,3 by using Theorem 3.2 that per(h 4,3 ) = per = 2c 1 c 2 + c 3 1 = f 1 2,3. c 1 ic 2 0 i c 1 ic 2 0 i c 1 Theorem 3.4 Let f2,n 1 be the first sequence of 2SO2F and L n = (l ij ) be an n n lower Hessenberg matrix given by l ij = c i j+1 c (i j) 2, if 0 i j < 2, 0, otherwise, that is Then, where c 0 = 1. L n = c 1 c c 1 c c c 1 c c 1 per(l n ) = f 1 2,n,. Proof. This is similar to the proof of Theorem 2.2 by using Theorem 3.1. Corollary 3.5 If we rewrite Theorem 3.2 and Theorem 3.4 for c i = 1, we obtain per(h n ) = F n+1 and per(l n ) = F n+1, respectively, where F n s are the Fibonacci numbers. Corollary 3.6 If we rewrite Theorem 3.2 and Theorem 3.4 for c 1 = 2 and c 2 = 1, we obtain per(h n ) = P n+1 and per(l n ) = P n+1, respectively, where P n s are the Pell numbers. Corollary 3.7 If we rewrite Theorem 3.2 and Theorem 3.4 with c 1 = 1 and c 2 = 2, then we obtain per(h n ) = J n+1 and per(l n ) = J n+1, respectively, where J n s are the Jacobsthal numbers.

8 Determinant and Permanent of Hessenberg Matrix Binet s formula for 2 sequences of generalized order 2 Fibonacci numbers (2SO2F) Let and n=0 a n z n be the power series of the analytical function f such that f(z) = a n z n when f(0) 0 n=0 A n = a 1 a a 2 a 1 a 0 0 a 3 a 2 a a n a n 1 a n 2 a 1 n n Then, the reciprocal of f(z) can be written in the following form g(z) = 1 f(z) = ( 1) n det(a n )z n, n=0 whose radius of converge is inf{ λ : f(λ) = 0}, [1]. Let Then, the reciprocal of p k (z) is where A k,n = p k (z) = 1 + a 1 z + + a k z k. (9) 1 p k (z) = ( 1) n det(a k,n )z n, Inselberg [4] showed that det(a k,n ) = n=0 a a 2 a a 3 a 2 a a k a k 1 a k a k a k a k a 1 k j=1 1 p k (λ j) n n. [8]. (10) ( ) n+1 1 (n k) (11) λ j

9 40 Kenan Kaygısız et al. if p k (z) has distinct zeros λ j for j {1, 2,..., k}; where p k(z) is the derivative of polynomial p k (z) in (9). Theorem 3.8 Let f 1 2,n be the first sequence of 2SO2F. Then, for n 2 and (c 1 ) 2 + 4c 1 c 2 > 0, f 1 2,n = k j=1 1 p (λ j ) ( ) n+1 1, (12) where p(z) = 1+c 1 z c 2 z 2 and p (z) denotes the derivative of polynomial p(z). Proof. This is immediate from Theorems 2.4 and (11). Corollary 3.9 Let f 2 2,n be the second sequences of 2SO2F. Then, for n 2. f 2 2,n+1 = c 2. k j=1 λ j 1 ( ) n+1 1 p (λ j ) λ j Proof. One can easily obtain the proof from (2) and Theorem 3.8. Acknowledgements The authors would like to express their pleasure to the anonymous reviewer for his/her careful reading and making some useful comments, which improved the presentation of the paper. References [1] E.T. Bell, Euler algebra, Trans. Amer. Math. Soc., 25(1923), [2] N.D. Cahill, J.R. D Errico, D.A. Narayan and J.Y. Narayan, Fibonacci determinants, College Math. J., 33(3) (2002), [3] M.C. Er, Sums of Fibonacci numbers by matrix method, Fibonacci Quart., 23(3) (1984), [4] A. Insenberg, On determinants of Toeplitz-Hessenberg matrices arising in power series, J. Math. Anal. Appl., 63(1978), [5] G.-Y. Lee and S.-G. Lee, A note on generalized Fibonacci numbers, Fibonacci Quart., 33(1995),

10 Determinant and Permanent of Hessenberg Matrix 41 [6] G.-Y. Lee, k-lucas numbers and associated bipartite graphs, Lineer Algebra Appl., 320(2000), [7] H. Minc, Encyclopaedia of Mathematics and its Applications, Permanents, (Vol.6), Addison-Wesley Publishing Company, London, (1978). [8] A.A. Öcal, N. Tuğlu and E. Altınışık, On the representation of k-generalized Fibonacci and Lucas numbers, Appl. Math. Comput., 170(2005), [9] F. Yılmaz and D. Bozkurt, Hessenberg matrices and the Pell and Perrin numbers, J. Number Theory., 131(2011),

arxiv: v1 [math.nt] 17 Nov 2011

arxiv: v1 [math.nt] 17 Nov 2011 On the representation of k sequences of generalized order-k numbers arxiv:11114057v1 [mathnt] 17 Nov 2011 Kenan Kaygisiz a,, Adem Sahin a a Department of Mathematics, Faculty of Arts Sciences, Gaziosmanpaşa