Global Journal of Mathematical Analysis, 5 () (07) -5 Global Journal of Mathematical Analysis Website: www.sciencepubco.com/index.php/gjma doi: 0.449/gjma.v5i.6949 Research paper On Generalized k-fibonacci Seuence by Two-Cross-Two Matrix Arfat Ahmad Wani *, G.P.S. Rathore Kiran Sisodiya 3 School of Studies in Mathematics, Vikram University, Ujjain, India epartment of Mathematics, Horticulture College, Msaur, India 3 School of Studies in Mathematics, Vikram University, Ujjain, India * Corresponding author E-mail:arfatahmadwani@gmail.com Abstract In this study we define a new generalized k-fibonacci seuence associated with its two cross two matrix called generating matrix. After use the matrix representation we find many interesting properties such as nth power of the matrix, Cassini s Identity of generalized k-fibonacci seuence as well as Binet s formula for generalized k-fibonacci seuence by the method of matrix diagonalization. Keywords: k-fibonacci Seuence; Generalized k-fibonacci Seuence; Binet s Formula; iagonalization of a Matrix.. Introduction The Fibonacci numbers have many interestiong properties applications to almost every fields of science art. For their amazing properties applications one can consult 0, 7, 8, 9. The beauty of Fibonacci numbers is that they can be generalize. So these numbers can be generalized by a number of ways these generalized forms have many interesting properties just like usual Fibonacci numbers. Many kinds of generalizations of these numbers have been presented in 3, 6, 7, 5. The two most important generalizations of Fibonacci numbers are k-fibonacci numbers {F k,n } k-lucas numbers {L k,n } these are defined as efinition. For any integer k, the kth Fibonacci seuence, say {F k,n } is defined recurrently by: F n+ kf k,n + F k,n with n 0 F k,0 0,F k, (.) efinition. For any integer k, the kth Lucas seuence, say {L k,n } is defined recurrently by: L n+ kl k,n + L k,n with n 0 L k,0,f k, k (.) The particular cases of definition (.) are If k, we obtain the classical Fibonacci seuence {0,,,,3,5...} If k, we obtain the Pell seuence {0,,,5,,9...} Many properties of k-fibonacci numbers obtained directly by matrix algebra in 6. Authors presented many interesting properties of k-fibonacci numbers in 5,. In 3 authors defined k-fibonacci numbers by using arithmetic indexes. Wloch in 4 discussed some identities for the generalized Fibonacci numbers the generalized Lucas numbers. In 4 author discussed some theorems or identities on the k-lucas numbers. Many properties of Fibonacci numbers as well as their generalizations have obtained in terms of matrices. In 8 authors studied the generalized Fibonacci Lucas numbers by matrix methods here the authors considered two cross two matrices after that obtained nth power of the matrices such as p if U(p,) 0 p p V (p,) p then U n Un+ U n (p,) U n U n ( p 4 ) n Un+ U n i f n is even V n U n U n (p,) ( p 4 ) n Vn+ V n i f n is odd V n V n where U n V n are nth generalized Fibonacci Lucas numbers respectively the authors defined these seuences recurrently by U n pu n U n, n, U 0 0, U V n pv n V n, n, V 0, V p In 0 authors considered two cross two matrix for U n V n in a different way derived a number of results by using this matrix which is defined below p p A In 7 author derived a number general formulas for the generalized Fibonacci seuence by matrix methods. Ahmet in obtained some identities of Pell, Pell-Lucas, Modified Pell numbers by using Copyright 07 Author. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited.
Global Journal of Mathematical Analysis some matrix methods. Here the authors defined some two cross two matrices as 3 N 6,R F In authors derived a number of properties of k-fibonacci k-lucas seuences with the help of two cross two generating matrix for these seuences, such as they proved a Binet s fomula for k- Fibonacci seuence k-lucas seuence by using the concept of diagonalization of generating matrix. So the generating matrix its nth powers are given as 0 F L k L k,n kl k,n L n k + 4 L k,n + L k,n+ k + 4 Fk,n then F n F k,n+ F k,n F k,n L k,n + L k,n+ k + 4 L k,n + L k,n+ k + 4 In author used the same concept as in studied the k-pell- Lucas seuences by matrix methods.. Generalized k-fibonacci Seuence In the present study we find the properties of generalized k-fibonacci seuence by matrix methods the generalized k-fibonacci is defined by efinition 3. For, k N, the generalized k-fibonacci seuence, say is defined recurrently by: S n k + with n S k,0,s k, k (.) The k-fibonacci seuence, k-lucas seuence generalized k- Fibonacci seuence have the same characteristic euation x kx. Let r s the two roots of this euation. Some conspicuous points about r s are r + s k, rs, k + 4, r kr s ks (.) where r k+ k +4 s k k +4 In 4 the well-known general forms for the k-fibonacci k- Lucas seuence are known as Binets formulae are given below F k,n rn s n, L k,n r n + s n (.3) the Binet s formula for the generalized k-fibonacci seuence (.) is given by rn+ s n+ Theorem. For k N, we have (.4) 3. Generating Matrix for the Generalized k- Fibonacci Seuence One of the most conventional methods for the study of the recurrences relations is generating matrix of the recurrence relations we are aware about that Fibonacci numbers their generalizations are the good examples of second order recurrence relations. But in the ongoing paper we concern about the generalized k-fibonacci seuence, so generalized k-fibonacci seuence is defined recursively as a linear combination of the p terms a n+p c p a n+p + c p a n+p + + c a n+ + c 0 a n (3.) where c 0,c c p are real constants for detailed illustration about the generating matrix one can see. If we put p in (3.) we get a n+ c a n+ + c 0 a n after that if we recall recurrence (.) take c 0 c k then the matrix associated called generating matrix is given by k S (3.) 0 Clearly S n ( ) n. For the nth power of S we have the following result Theorem. For k N, we have S n Sk,n, n (3.3) Proof. To prove the result we will use induction on n. Clearly (3.3) is true for n. Suppose (3.3) is true for n, we get S n+ S n S Sk,n k 0 ksk,n + k + Sk,n+ Corollary. For k,n N, we have S n Fk,n+ F k,n F k,n F k,n Theorem 3. (Cassini s Identity) For k,n N, we have (3.4) + S k,n ( ) n+ (3.5) Proof. It can be simply proved by using the concept of determinats to matrices S S n in euations (3.) (3.3). Proof. Here we can employ cramers rule in 9 for linear systems of euations to derive Cassini s identity. Consider a linear system a + b + + a + b + (3.6) L k,n +, n (.5) L k,n (k + 4) S k,n 4 ( ) n, n (.6) Proof. It can be simply prove by the use of euations (.),(.3) (.4) Clearly Sk,n + 0 for n. Let Sk,n + then by Cramer s rule, we get + + S k,n+ a b + + +
Global Journal of Mathematical Analysis 3 By virtue of the recurrence relation (.), a k b is the uniue solution of the system given in (3.6). Therefore by Cramer s rule, we get + + S k,n+ S k,n+ Sk,n+ Sk,n + Let P k,n + S k,n (3.7) P k,n+ + S k,n+ Clearly P k,n+ P k,n with n,p k, (3.8) Now the euation (3.8) is a first order linear recurrence. Thus P K,n ( ) n+ is the general solution of (3.8). Hence from the euation (3.7), we get + S k,n ( ) n+ Theorem 4. For k,n N, we have Sk,n+ Sk,n S (3.9) Proof. To prove the result we will use induction on n. (3.9) is true for n. Suppose (3.9) is true for n, we get Sk,n+ ksk,n+ + + + k Sk,n+ 0 k k Sk,n 0 0 k ksk,n + 0 k Sk,n+ 0 Sk,n+ S Theorem 5. For k,n N, we have Sk,n+ S n Sk, S k,0 (3.0) Proof. It can be show simply by Principal of Mathematical Induction. 4. Binet s Formula by Matrix iagonalization of Generating Matrix In this section we will use the diagonalization of the generating matrix to obtain the Binet s fomula for the generalized k-fibonacci seuence defined in (.). Theorem 6. (Binet s Formula): For n 0 k N, the nth term of the generalized k-fibonacci seuence is given by rn+ s n+ (4.) where r s are the roots of the characteristic euation x kx 0 k Proof. Since the generating matrix is given by S 0. Now here we have motive to diagonalize the generating matrix S. Since S is a suare matrix. So let x be the eigen value of U then by the Cayley Hamilton theorem on matrices, we have U xi 0 k x x x 0 x kx x 0 (4.) This is the characteristic euation of the generating matrix. Let r k+ k +4 s k k +4 are the roots of the characteristic euation also r s be the two eigen values of a suare matrix S. Now we will try to find the eigen vectors corresponding to the eigen values r s. To find the eigen vectors we simply solve the system of linear euations given by (S xi)v 0 where V is the column vector of order. First of all we calculate the eigen vector corresponding to the eigen value r then (S ri)v 0 k r r V r V 0 (k r)v + rv 0 (4.3) V rv 0 (4.4) put V t in (4.4) we get V rt. Hence the eigen vectors corresponding to r are. In particular t, the eigen vector rt t r corresponding to r is s Similarly the eigen vector corressponding to s is. Let A r s be the matrix of eigen vectors, so P then A s (). Now we consider a diagonal matrix r in which eigen values of S are on the main diagonal, By the principal of diagonalization of matrices, we have S AA S n (AA ) n A n A () r s () r n+ s n+ r n s n r 0 0 s r n 0 s 0 s n r s () r n+ s n+ sr n+ + rs n+ r n s n sr n + rs n Since from euation (.) rs ( ), we have r S n () n+ s n+ r n s n Sk,n+ Since S n Sk, S k,0 S n k r r n s n r n s n S n k then.
4 Global Journal of Mathematical Analysis Sk,n+ r () n+ s n+ r n s n k r n s n r n s n kr () n+ ks n+ + r n s n kr n ks n + r n s n () r n (kr + ) s n (ks + ) r n (kr + ) s n (ks + ) After using euation (.), we have Sk,n+ r n+ s n+ r n+ s n+ Hence rn+ s n+ This is clearly the Binet s formula for generalized k-fibonacci seuence which is defined in euation (.4) Theorem 7. The generalized characteristic roots of S n are r n L k,n + k + 4 s n L k,n k + 4 Proof. If we write the characteristic polynomial of S n, we have S n yi S k,n y S k,n y Sk,n y y ( y )( y ) S k,n y ( ) + y + Sk,n If we recall euations (.5) (3.5), we have S n yi y L k,n y + ( ) n y L k,n y + ( ) n Thus the characteristic euation of S n is y L k,n y + ( ) n 0 the generalized characteristic roots are given by L k,n ± Lk,n 4( )n y If we use euation (.6) in euation (4.7), we get y L k,n ± k + 4 (4.5) (4.6) (4.7) (4.8) Clearly the euation (4.8) has two roots these are r n s n. Now accordingly we get the desired result as r n L k,n + k + 4 Since s n L k,n k + 4 S n Sk,n S n S k,n Since the ratio of the two consecutive generalized k-fibonacci numbers is eual to r that is Therefore lim r n lim lim n n r S n lim r r n r If we consult euation (.), we have S n lim r r kr + r n r r If we compute the determinants of both sides, we get the characteristic euation of the S matrix as below 5. Conclusion (kr + r ) 0 r kr 0 In the present study we obtained nth power of the matrix some properties have been obtained for the generalized k-fibonacci seuence by matrix methods. Acknowledgement we would like to thank anonymous referee for cautiously reading the paper for their remarks which decently upgraded the paper. References A. Borges, P. Catarino, A. P. Aires, P. Vasco H. Campos. Two-by- Two Matrices Involving k-fibonacci k-lucas Seuences, Applied Mathematical Sciences, 8(34):659 666, 04. A. asdemir. On the Pell, Pell-Lucas Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences, 5(64):373 38, 0. 3 A. F. Horadam. Generalized Fibonacci Seuences, The American Mathematical Monthly, 68(5):455-459, 98. 4 A. Wloch. Some identities for the Generalized Fibonacci numbers the Generalized Lucas numbers, Applied Mathematics Computation, 9:5564 5568, 03. 5 C. Bolat. On the Properties of k-fibonacci numbers. Int. J. contemp. Math. Sciences, 5():097 05, 00. 6 C. K. Ho. C. Y. Chong. Odd Even Sums of Generalized Fibonacci Numbers by Matrix Methods, AIP Conference Proceedings, 60, 06 (04); doi: 0.063/.48860 7. Kalman. Generalized Fibonacci numbers by Matrix Methods, The Fibonacci Quarterly, 0():73 76, 98. 8 G. C. Morales. On Generalized Fibonacci Lucas Numbers by Matrix Methods, Hacettepe Journal of Mathematics Statistics, 4():73 79, 03. 9 http://mathworld.wolfram.com/cramersrule.html. 0 N. N. Vorobyov. The Fibonacci Numbers,. C. Health company Boston, 963. P. Catarino. A Note Involving Two-by-Two Matrices of the k-pell k-pell-lucas Seuences, International Mathematical Forum, 8(3):56 568, 03. P. Catarino. On Some Identities for k-fibonacci Seuence, Int. J. contemp. Math. Sciences. 9():37 4, 04. 3 S. Falcon A. Plaza. On k-fibonacci numbers of arithmetic indexes, Applied Mathematics Computation,08:80 85, 009. 4 S. Falcon. On the k-lucas Numbers, Int. J. contemp. Math. Sciences. 6():039 050, 0.
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