Balancing sequences of matrices with application to algebra of balancing numbers

Transcription

1 Notes on Number Theory and Discrete Mathematics ISSN Vol No Balancing sequences of matrices with application to algebra of balancing numbers Prasanta Kumar Ray International Institute of Information Technology Bhubaneswar India prasanta@iiit-bhacin Abstract: It is well known that the problem of finding a sequence of real numbers a n n = which is both geometric a n+1 = ka n n = and balancing a n+1 = 6a n a n 1 a 0 = 0 a 1 = 1 admits an unique solution In fact the sequence is 1 λ 1 λ 2 1 λ n 1 where λ 1 = satisfies the balancing equation λ 2 6λ + 1 = 0 In this paper we pose an equivalent problem for a sequence of real nonsingular matrices of order two and show that this problem admits an infinity of solutions that is there exist infinitely many such sequences Keywords: Balancing numbers Lucas-balancing numbers Balancing matrix Lucas-balancing matrix AMS Classification: 11B39 11B83 1 Introduction Behera and Panda [1] recently introduced a number sequence called balancing numbers defined in the following way: A positive integer n is called a balancing number with balancer r if it is the solution of the Diophantine equation n 1 = n n n + r By slightly modifying this Diophantine equation Panda and Ray [6] introduced cobalancing numbers as the solutions of the Diophantine equation n = n n n + R 49

2 where R is the cobalancer corresponding to the cobalancer n Balancing numbers B n and the related Lucas-balancing numbers C n = 8n are defined recursively by the formulas B n+1 = 6B n B n 1 with B 0 = 0 B 1 = 1 and C n+1 = 6C n C n 1 with C 0 = 1 C 1 = 3 [7 8] Both the relations satisfy the balancing equation λ 2 6λ + 1 = 0 11 which admits an unique solution λ 1 λ 2 with λ 1 = and λ 2 = 3 8 In fact a sequence of balancing constants 1 λ 1 λ 2 1 λ n 1 12 satisfies the balancing equation 11 where the balancing mean lim n B n+1 B n = λ 1 = Many properties of balancing numbers and their related sequences are available in the literature Interested readers can follow [ ] In this paper we pose an equivalent problem for a sequence of real non-singular matrices of order two and we will show that this problem admits an infinitely many such sequences of balancing constants Indeed each sequence is naturally related to each other in terms of the generator of the sequences Finally with the help of some basic tools from the theory of matrices to the generators of these balancing sequences we deduce some of the more familiar balancing and Lucas-balancing identities and the fascinating Binet s formulas for the general terms of balancing and Lucas-balancing numbers 2 Defining equations Consider the matrix x A = u y v and the identity matrix 1 0 I = 0 1 where the entries x y u v are to be determined subject to the constraint xv yu 0 Notice that the geometric sequence I A A 2 A n is to be balancing if and only if it will satisfy the balancing equation A 2 = 6A I 21 that is x u y x v u y x = 6 v u y v 0 1 It is clear that for A is not nilpotent Since we are restricted A to be nonsingular 21 implies to the following identity: A n+1 = 6A n A n 1 n =

3 Further 22 is equivalent to the following system of equations: x 2 6x yu = 0 23 x + v 6y = 0 24 x + v 6u = 0 25 v 2 6v yu = 0 26 Now consider the following cases to investigate possible solution sets of the mentioned equations First we consider y = 0 and observe that 23 and 26 reduce to balancing equation 11 giving the solution sets x = {λ 1 λ 2 } where λ 1 = λ 2 = 3 8 Further if u = 0 solution matrices of 22 are λ 1 0 λ 1 0 λ 2 0 λ 2 0 Φ 0 = Φ 1 = Φ 2 = Φ 3 = 0 λ 2 0 λ 1 0 λ 1 0 λ 2 Notice that the following correlations for the balancing constants λ 1 λ 2 every n are valid for λ 1 + λ 2 = 6 λ 1 λ 2 = 2 8 λ 1 λ 2 = 1 λ 2 1 = 6λ 1 1 λ 2 2 = 6λ 2 1 λ n+1 1 = 6λ n 1 λ n 1 1 λ n+1 2 = 6λ n 2 λ n 1 2 Applying appropriate identities we observe that each of the sequences {Φ n 0} {Φ n 1} {Φ n 2} {Φ n 3} is a balancing sequence 12 Again for u 0 25 reduces to x +v 6 = 0 and therefore the solution matrices of 22 are given by λ 1 0 λ 2 0 Φ 0u = Φ 2u = u λ2 u λ1 It can be easily shown that the general term of the sequence generated by Φ 0u will be where B n is the n th balancing number Φ n λ n 1 0 0u = B n u λ n 2 Consider the next case when y 0 In this case if u = 0 23 and 26 reduce to the balancing equation 11 and 25 implies x + v = 6 The situation is similar to the previous case when u 0 Further when u 0 25 reduces to x = 6 v consistent with 24 Substituting the value of x in 23 we get 6 v 2 66 v yu = 0 51

4 Further simplification gives the equation v 2 6v yu = 0 consistent with 26 Thus assuming y 0 u 0 23 to 26 reduce to the following equivalent system: v = 3 ± 8 yu x = 6 v 27 Indeed this system of equations will help to investigate various sets of solutions of 22 which will be discussed in the following section 3 Some examples of balancing sequences Example 31 Setting y and u to integer values in 27 we observe that there is a unique pair y = 1 u = 1 which keeps the radicand positive In this case we obtain two sets of solutions: x = 0 v = 6 y = 1 u = 1; and x = 6 v = 0 y = 1 u = 1 The latter set results the balancing matrix Q B = sequence is where Q n B = B n+1 B n B n B n 1 I Q B Q 2 B Q 3 B Q n B [12] 6 1 and the corresponding balancing 1 0 Example 32 Now for the pair y = u = 0 in 27 results in the matrix Φ 0 and the corresponding balancing sequence will be I Φ 0 Φ 2 0 Φ n 0 where Φ n 0 = λ n λ n 2 The roots of the characteristic equation balancing equation 11 of Q B are indeed λ 1 and λ 2 Observe that these roots are nothing but the diagonal elements of Φ 0 Thus 11 is the characteristic equation for both Q B and Φ 0 and by Cayley-Hamilton theorem each of these matrices satisfy this equation Comparing 21 and 11 we have in fact a characterization for all matrices which give rise to balancing sequence of the type 12 The following theorem demonstrates the fact: Theorem 33 A necessary and sufficient condition for the matrix A to be a generator of a balancing sequence 12 is that its characteristic equation is the balancing equation The following corollary is an immediate consequences of Theorem 33 52

5 Corollary 34 Any two matrix generators of non-trivial balancing sequence of matrices are similar The following corollary immediately follows from Corollary 34 Corollary 35 The matrices Q B and Φ 0 are similar that is there exist a non-singular matrix T such that Q B = T Φ 0 T 1 where the columns of T are eigenvectors of Q B corresponding to the eigenvalues λ 1 and λ 2 Since Q B T = T Φ 0 a straight forward computation shows that 1 1 T = λ 1 λ 2 It is clear from Corollary 34 that Q n B = T Φn 0T 1 and hence Q n B is similar to Φn 0 Thus detq n B = detφ 0 ; traceq n B = traceφ 0 and we obtain our first pair of balancing identities as follows Corollary 36 For every natural number n B 2 n B n+1 B n 1 = 1; B n+1 B n 1 = λ n 1 + λ n 2 Since the identity B n+1 B n 1 = 2C n is valid for all natural number n [8] we notice that the second part of Corollary 36 establishes the proof of the Binet s formula for Lucas-balancing numbers C n = λ 1+λ 2 [7 8] 2 Further motivated from the general term Q n B = B n+1 B n in Example 31 we will B n B n 1 now examine that whether the sequence with general term P n C n+1 C n = is a balancing C n C n sequence 12 Observe that for n = 1 the matrix P = does not satisfy the balancing equation 11 Thus from Theorem 33 we conclude that P n is not a balancing sequence 3 1 Nevertheless we will show in the next example that the Lucas-balancing numbers do in fact enter into the picture in a natural way Example 37 Setting y = 1 u = 8 in 27 yields v = 3 x = 3 and we obtain the sequence 3 1 generator S = The corresponding balancing sequence is

6 The general term of the above sequence can be easily shown as S n = C n 8B n B n C n Similarity of Q n B with Sn by the invariance of trace results and by the invariance of determinant results 2C n = B n+1 B n 1 C 2 n 8B 2 n = B 2 n B n+1 B n 1 = 1 Whereas similarity of Φ n 0 with S n implies by trace gives rise to the Binet s formula for Lucasbalancing numbers C n = λn 1 +λn 2 2 and by the determinant λ n 1 λ n 2 = C 2 n 8B 2 n = 1 Example 38 Setting y = 1 u = 1 in 27 one set of solutions is v = 6 x = 0 and we obtain 0 1 the matrix R = The general form will be 1 6 R n = B n 1 B n B n B n+1 Similarity of R n with Φ n 0 and Q n B gives the results of Corollary 36 whereas similarity with Sn gives the same results as in Example 37 Example 39 By taking y = 1 u = 1 in 27 we obtain v = 1 x = 7 and the generator F n 7 1 = The corresponding balancing sequence is where the general term can be easily shown as L n = c n 8B n B n c n 1 where b n denotes the n th cobalancing number and c n = 8b 2 n + 8b n + 1 denotes the n th Lucascobalancing number [6 7 8] 54

7 Similarity with Q n B Φn 0 S n respectively give rise to the following identities c n c n 1 B n+1 B n 1 = 7Bn 2 8Bn 2 c n c n 1 = 1 Cn 2 + c n c n 1 = 16Bn 2 Whereas similarity with R n gives same result as Q n B 4 A matrix approach for the proof of Binet s formula for balancing and Lucas-balancing numbers In this section we offer two examples of generators of balancing sequences and compute their eigenvalues With this new tool we will then establish the Binet s formula for B n and C n in terms of λ 1 and λ 2 Choose u = 0 and y 0 to be arbitrary in 27 we obtain v = λ 1 and x = λ 2 and the matrix λ 1 y Φ 0y = Φysay = 0 λ 2 The corresponding balancing sequence is 1 0 λ 2 1 6y λ 2 2 λ y 0 λ 3 2 The general term of the above sequence can be easily seen as Φ n λ n 1 B n y y = 0 λ n 2 The eigenvectors of the matrix Φ y corresponding to the eigenvalues λ 1 and λ 2 are respectively a a and λ 0 2 λ 1 where a 0 Choose a = 1 Since λ 2 λ 1 = 2 8 and taking y = a y we have two eigenvectors and corresponding to the eigenvalues λ 1 and λ respectively Setting D = and using Corollary 34 we obtain Φ = DΦ 0 D 1 It follows that Φ n 2 = 8 DΦn 0D 1 and hence Φ n 2 8 D = DΦn

8 Now 41 can be rewritten as λ n 1 2 8λ n 1 = 0 λ n 2 λ n 1 λ n 2 0 λ n 2 Equating the elements of first row and second column we get B n = λ 1 λ which is nothing but the Binet s formula for balancing numbers Finally we present an example where we permit our generator matrix to be complex Setting y = 3 u = 3 in 27 we obtain v = 3 + i x = 3 i and the matrix E = The corresponding balancing sequence is i i 3 i i 17 6i i with the general term E n C n B n i = 3B n 3B n C n + B n i Proceeding as in the previous example we obtain the eigenvectors corresponding to the eigenvalues λ 1 and λ 2 respectively as [λ i] 1 1 and [λ i] We construct the 1 1 matrix 1 M = [λ i] [λ i] 1 1 and using Corollary 34 we obtain E n M = MΦ n 0 which on simplification and comparison of second row and first column entries of the matrix gives B n [λ i] + C n + ib n = λ n 1 Further simplification results λ n 1 = λ 1 3B n + C n that is λ n 1 = 8B n + C n 42 56

9 Similarly we can obtain λ n 2 = 8B n + C n 43 Adding 42 and43 we obtain C n = λ 1 + λ 2 2 which is the Binet s formula of Lucas-balancing numbers References [1] Behera A G K Panda On the square roots of triangular numbers The Fibonacci Quarterly Vol No [2] Berczes A K Liptai I Pink On generalized balancing numbers Fibonacci Quarterly Vol No [3] Keskin R O Karaatly Some new properties of balancing numbers and square triangular numbers Journal of Integer Sequences Vol No 1 Art #1214 [4] Liptai K F Luca A Pinter L Szalay Generalized balancing numbers Indagationes Math N S Vol [5] Olajos P Properties of balancing cobalancing and generalized balancing numbers Annales Mathematicae et Informaticae Vol [6] Panda G K P K Ray Cobalancing numbers and cobalancers International Journal of Mathematics and Mathematical Sciences Vol No [7] Panda G K P K Ray Some links of balancing and cobalancing numbers with Pell and associated Pell numbers Bulletin of the Institute of Mathematics Academia Sinica New Series Vol No [8] Panda G K Some fascinating properties of balancing numbers Proc Eleventh Internat Conference on Fibonacci Numbers and Their Applications Cong Numerantium Vol [9] Ray P K Application of Chybeshev polynomials in factorization of balancing and Lucasbalancing numbers Bol Soc Paran Mat Vol No [10] Ray P K Factorization of negatively subscripted balancing and Lucas-balancing numbers Bol Soc Paran Mat Vol No [11] Ray P K Curious congruences for balancing numbers Int J Contemp Sciences Vol No

10 [12] Ray P K Certain matrices associated with balancing and Lucas-balancing numbers Matematika Vol No [13] Ray P K New identities for the common factors of balancing and Lucas-balancing numbers International Journal of Pure and Applied Mathematics Vol No [14] Ray P K Some congruences for balancing and Lucas-balancing numbers and their applications Integers Vol #A8 [15] Ray P K On the properties of Lucas-balancing numbers by matrix method Sigmae Alfenas Vol No