In this thesis, we have studied the balancing and Lucas-balancing numbers for real indices. Also we have discussed some properties of balancing numbers with real numbers. We extend some properties of Fibonacci numbers to balancing numbers

Tanty, Sephali

ethesis@nitr

Balancing And Lucas-balancingNumbers With Real IndicesA thesissubmitted bySEPHALI TANTYRoll No. 413MA2076forthe partial fulfilment for the award of the degreeMaster Of ScienceUnder the supervision ofProf. GOPAL KRISHNA PANDADEPARTMENT OF MATHEMATICSNATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA, 769008c© Sephali TantyMay 2015DECLARATIONI here by declare that the topic “Balancing and Lucas balancing numbers for realindices” as a partial fulfillment of my M.Sc. degree has not been submitted to anyother institution or the university for the award of any other degree or diploma.Name:Sephali TantyRoll No:413MA2076Dept. of MathematicsNIT, RourkelaNATIONAL INSTITUTE OF TECHNOLOGY, ROURKELACERTIFICATEThis is to certify that the project report entitled Balancing and Lucas-balancingnumbers with real indices submitted by Sephali Tanty to the National Insti-tute of Technology, Rourkela, Odisha for the partial fulfilment of requirements forthe degree of Master of Science in Mathematics is a bonafide record of work carriedout by her under my supervision and guidance. The contents of the project, in fullor in parts have not been submitted to any other institution or the university or theaward of any other Degree or Diploma.Prof. Gopal Krishna PandaSupervisorDepartment of MathematicsNational Institute of TechnologyRourkela (India) -769008May 2015ACKNOWLEDGEMENTI want to express my gratefulness to my supervisor Prof. Gopal Krishna Panda,Department of Mathematics, National Institute of Technology, Rourkela, whose in-spiring guidance made this project possible.I also like to thank Prof. S. Chakraverty, Head of Mathematics Department, NITRourkela for providing departmental computing facilities.I further like to thank all the faculty members and Ph.D. scholars Mr. SudhansuSekhar Rout and Mr. Ravi Kumar Davala for their support during the preparationof this work.I am very grateful to the Director of NIT, Rourkela, Prof. Sunil Kumar Sarangi,for providing excellent facilities for research in the institute.Lastly, I want to thank my parents for their untiring support for me.Sephali TantyRoll Number: 413MA2076ABSTRACTIn this thesis, we have studied the balancing and Lucas-balancing numbers forreal indices. Also we have discussed some properties of balancing numbers with realnumbers. We extend some properties of Fibonacci numbers to balancing numbers.Contents1 INTRODUCTION 62 Preliminaries 82.1 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Fibonacci sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Lucas sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Binet Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Pell’s numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Associated Pell’s numbers . . . . . . . . . . . . . . . . . . . . . . . . 102.8 Balancing Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Lucas-Balancing Numbers . . . . . . . . . . . . . . . . . . . . . . . . 113 Balancing and Lucas-balancing with real indices 123.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Fibonacci and Lucas numbers with real indices . . . . . . . . . . . . . 133.3 Balancing and Lucas-balancing numbers with real indices . . . . . . 133.4 Some Properties of Fibonacci and Lucas numbers . . . . . . . . . . . 143.5 Some Properties of balancing and Lucas-balancing numbers . . . . . . 153.6 Proofs of the Properties for balancing and Lucas-balancing numbers . 165CHAPTER-11. INTRODUCTIONThere is a famous quote by the famous Johann Carl Friedrich Gauss a GermanMathematician (30 April, 1777 − 23 February, 1855) : “Mathematics is the queen ofall Sciences, and Number theory is the queen of Mathematics”, (see [4, 5]). Numbertheory, or higher arithmetics is the study of those properties of integers which weused everyday arithmetic.In Number theory (see [4, 5]), discovery of number sequences with certain specifiedproperties has been a source of attraction since ancient times. The most beauti-ful and simplest of all number sequences is the Fibonacci sequence. This sequencewas first invented by Leonardo of Pisa (1170 − 1250), who was also known as Fi-bonacci, to describe the growth of a rabbit population. The Fibonacci numbers are1, 1, 2, 3, 5, 8, 13, · · · and denoted by Fn, n ≥ 1.Generally known, Number Theory as the study of properties of numbers. But now-a-days the application of Number Theory uses in diverse fields like, Coding Theory,Cryptology (the art of creating and breaking codes), Computer Science.Other interesting number sequences are Pell’s sequence and the Associated Pell’ssequence, (see [4, 5]). In Mathematics, the Pell’s numbers are infinite sequence ofintegers that have been known since ancient times. The denominators of the closestrational approximations to the square root of 2. This sequence begins with 11,32,75,1712,4129;so the sequences of Pell’s numbers begin with 1, 2, 5, 12, 29. The numerators of thesame sequence of approximations give the Associated Pell sequence.6The concept of balancing numbers was first invented by Behera and Panda, (see[1]) in the year 1999 in connection with a Diophantine equation. It consists of findinga natural number n such that:1 + 2 + 3 + · · ·+ (n− 1) = (n+ 1) + (n+ 2) + · · ·+ (n+ r)for some natural number r, is balancer corresponding to the balancing number n. Ifthe nth triangular number n(n+1)2is denoted by Tn, then the above equations reducesto Tn−1+Tn = Tn+r, which is the problem finding two consecutive triangular numberswhose sum is also a triangular number. Since,T5 + T6 = 15 + 21 = 36 = T86 is a balancing number with balancer 2. Similarly,T34 + T35 = 595 + 630 = 1225 = T49implies that, 35 is also a balancing number with balancer 14. The balancing numbers,though obtained from a simple Diophantine equation, are very useful for the compu-tation of square triangular numbers. An important result about balancing numbersis that, n is a balancing number if and only if 8n2 + 1 is a perfect square, and thenumber√8n2 + 1 is called a Lucas- balancing numbers is that, these numbers areassociated with balancing numbers in the way Lucas numbers are associated withFibonacci numbers, (see [1, 2]).There are some other properties common to both Fibonacci as well as balancingnumbers. As shown by Behera and Panda, (see [1, 10]), the square of balancingnumbers is is a triangular number.The search for balancing numbers in well knowninteger sequences was first initiated by Liptai, (see [2]). He proved that, there is nobalancing number in the Fibonacci sequences other than 1.7CHAPTER-22. PreliminariesIn this chapter, we recall some definitions known results of Fibonacci and Lucasnumbers, Balancing and Lucas-balancing numbers, Pell’s numbers, Associated Pell’snumbers, Real numbers, Recurrence relations, Binet formula, Diophantine equations,complex function.2.1. Recurrence relationsIn Mathematics, a recurrence relations is an equation that defines a sequences re-cursively, each term of the sequences is defined as a functions of the preceding terms,(see [4, 5]).2.2. Fibonacci sequencesThe Fibonacci numbers are defined recursively as F1 = 1, F2 = 1 andFn+1 = Fn + Fn−1, n ≥ 2.(see [4, 5]).2.3. Lucas sequencesLucas sequences is also obtained from the same recurrence relation as that forFibonacci numbers. The Lucas numbers are defined recursively as L1 = 1, L2 = 3andLn+1 = Ln + Ln−1, n ≥ 2.(see [4, 5]).82.4. Diophantine equationsIn Mathematics, a Diophantine equations is an intermediate polynomial equa-tions that allows the variables to be integers only. Diophantine problems have fewerequations than unknowns and involve finding integers that work correctly for all theequations, (see [4, 5, 9]).2.5. Binet FormulaWhile solving a recurrence relations, as a difference equations, the nth term of thesequence is obtained in closed form, which is a equation containing conjugate surdsof irrational number is known as the Binet Formula for the particular sequence. TheBinet Formula for the Fibonacci sequences is,Fn =αn1 − βn1α1 − β1 .The Binet Formula for the Lucas sequences is given by,Ln = αn1 + βn1 .where n ∈ N , α1 = 1+√52, β1 =1−√52,(see [1, 4, 5, 9]).2.6. Pell’s numbersThe Pell’s numbers are defined recursively as P1 = 1, P2 = 2 andPn = 2Pn−1 + Pn−2, n ≥ 2.(see [4, 5]),1. Cassini Formula: Pn−1Pn+1 − P 2n = (−1)n2. Binet Formula:αn2−βn2α2−β2 where, α2 = 1 +√2 and β2 = 1−√2,9andPn = 1, 2, 5, 12, 29, · · · , n = 1, 2, · · ·2.7. Associated Pell’s numbersThe Associated Pell’s numbers are defined recursively as Q1 = 1, Q2 = 2 andQn = 2Qn−1 +Qn−2, n ≥ 2.(see [4, 5, 10]),1. Cassini Formula: Qn−1Qn+1 −Q2n = (−1)n.2. Binet Formula:αn2−βn2α2+β2where, α2 = 1 +√2 and β2 = 1−√2,andQn = 1, 3, 7, 17, · · · , n = 1, 2, · · ·2.8. Balancing NumbersThe balancing numbers are defined recursively as B1 = 1, B2 = 6 andBn+1 = 6Bn −Bn−1, n ≥ 2.(see [1, 2, 9, 10]),1. Cassini Formula: B2n −Bn+1Bn−1 = 1.2. Binet Formula: αn−βnα−β where, α = 3 +√8 and β = 3−√8, andBn = 1, 6, 35, 204, 1189, · · · , n = 1, 2, · · ·102.9. Lucas-Balancing NumbersThe Lucas-balancing numbers are defined recursively as C1 = 3, C2 = 17 andCn+1 = 6Cn − Cn−1, n ≥ 2.(see [1, 2, 9, 10]),1. Cassini Formula: C2n − Cn+1Cn−1 = −8.2. Binet Formula: αn−βn2where, α = 3 +√8 and β = 3−√8, andCn = 3, 17, 99, 577, 3363 · · · , n = 1, 2, · · ·11CHAPTER-33. Balancing and Lucas-balancing with real indicesR. Witula, (see [12]) discovered some new formulas for Fibonacci and Lucasnumbers by using real numbers.3.1. IntroductionUsually, every number sequence is defined for integral indices. In a recent paper,(see [12]), Witula introduced Fibonacci and Lucas numbers with real indices andprovided some applications. We devote this chapter to explain the work of Witula,(see[12]).Using the Binet Formulas for Fibonacci and Lucas numbers, (see [1, 4, 5]), one caneasily get,1.√5Fn = αn1 − βn1 ,2. Fn+1 − β1Fn = αn1 ,3. Ln = 2αn1 −√5Fn.Using the Binet formulas of balancing and Lucas-balancing numbers the followingresults can be obtained.2√8Bn = αn − βn, (1)Cn = αn −√8Bn. (2)It is easy to see that,Bn+1 − βBn = αn. (3)where, α = 3 +√8 and β = 3−√8.123.2. Fibonacci and Lucas numbers with real indicesThese following results were developed by the R.Witula, (see [12]).Let s be real number. Then,1.√5Fs = αs1 − eipis(−βs1),2. Fs+1 = αs1 + β1Fs,3. Ls = 2αs1 +√5Fs.where α1, β1 > 0 and eipis = cospis+ i sin pis, is a complex function. Then,Fs+1 = αs1 + β1Fs=√5Fs + eipis(−β1)s + β1Fs.Hence,Fs+1 = eipis(−β1)s + α1Fs.3.3. Balancing and Lucas-balancing numbers with real indicesLet s be real number. We define balancing numbers Bs and Lucas-balancingnumbers Cs with real indices as,2√8Bs = αs − eipis(−β)s, (4)Cs = αs −√8Bs. (5)It is easy to see that,Bs+1 = αs + βBs. (6)where α, β > 0 and eipis is a complex function.13Note that, (4) and (6) imply,Bs+1 = αs + βBs= 2√8Bs + eipis(−β)s + βBs.Hence,Bs+1 = eipis(−β)s + αBs. (7)3.4. Some Properties of Fibonacci and Lucas numbersThe following are some properties related to Fs and Ls , s ∈ R, (see [12]).1.√5Fs = αs − eipis(−β)s.2. Ls = αs + eipis(−β)s.3. Fs+2 = Fs+1 + Fs.4. Ls+2 = Ls+1 + Ls.5. F−s = −e−ipisFs.6. L−s = e−ipisLs.7. F2s = FsLs.8. Ls = Fs+1 + Fs−1.9. 5F 2s = L2s − 2eipis.10. 5F 3s = F3s − 33eipisFs.11. Fs+t+1 = Fs+1Ft+1 − FsFt.12. eipitLs−t = Ft+1Ls − FtLs+1.13. 25F 4s = L4s − 4eipisL2s + 6e3ipis.14. FsFt − Fs−rFt+r = eipi(s−r)FrFt−s+r.143.5. Some Properties of balancing and Lucas-balancing numbersWe prove the following properties of Bs and Cs with s ∈ R.2Cs = αs + eipis(−β)s. (8)Bs+2 = 6Bs+1 −Bs. (9)Cs+2 = 6Cs+1 − Cs. (10)B−s = ±e−ipisBs = eipisBs, if s is odd;−eipisBs, if s is even. (11)C−s = ±e−ipisCs = −eipisCs, if s is odd;eipisCs, if s is even. (12)B2s = 2BsCs. (13)2Cs = Bs+1 −Bs−1. (14)± eipitCs−t = Bt+1Cs −BtCs+1 = eipit, if s is even;−eipit, if s is odd. (15)BsBt −Bs−rBt+r = eipi(s−r)BrBt−s+r. (16)Bs+t+1 = Bs+1Bt+1 −BsBt. (17)16B2s = C2s ± eipis = eipis, if s is odd;−eipis, if s is even. (18)32B3s = B3s ± 3eipisBs = 3eipisBs, if s is odd;−3eipisBs, if s is even. (19)153.6. Proofs of the Properties for balancing and Lucas-balancing numbersUsing equations (6), we get (9) as follows.Bs+2 = 6Bs+1 −Bs.Bs+2 = αs+1 + β(αs + βBs)= 6αs + β2Bs= 6αs + 6βBs −Bs= 6Bs+1 −Bs,The proof of (10) is similar to that of (9).Using (4) and the fact that αβ = 1, (11) we have,B−s = ±e−ipisBs.2√8B−s = α−s − e−ipis(−β)s= βs − e−ipis(−α)s= [eipis(β)s − (−1)sαs]/eipis= [(−1)2seipis(β)s − (−1)sαs]/eipis= (−1)s+1[αs − (−1)seipis(β)s]e−ipis= (−1)s+1[αs − eipis(β)s]e−ipis= (−1)s+12√8Bse−ipis,The proof of (12) is same as that of (11).16Putting the value of (4) and (8), we get (13) as follows.B2s = 2BsCs.2√8B2s = α2s − e2ipis(−β)s]= 2Cs[αs − (−1)seipis(β)s]= 2Cs2√8Bs.Using (5), we get (14) as follows.2Cs = Bs+1 −Bs−1.Cs = αs −√8Bs= [2αs + βBs − αBs]/2= [Bs+1 + α(αs−1 −Bs)]/2[∵ Bs = αs−1 +Bs−1]= [Bs+1 −Bs−1]/2.Putting the value of (4) and (7), we get (15) as follows.Fromeipitαs(−β)t + eipisαt(−β)s= eipitαt(−βt)[αs−t + eipi(s−t)(−βs−t)]= (−1)teipit(2Cs−t).17andeipitαs(−βt) + eipisαt(−βs)= eipit(−βt)(Bs+1 − βBs) + αt(Bs+1 − αBs)= (Bt+1 − αBt)(Bs+1 − βBs) + (Bt+1 − βBt)(Bs+1 − αBs)= Bt+1(2Bs+1 − 6Bs) +Bt(2Bs − 6Bs+1)= Bt+1(2Bs+1 −Bs+1 −Bs−1) +Bt(2Bs −Bs+2 −Bs)= Bt+12Cs −Bt2Cs+1.(15) follows.Using (4), we get (16) as follows.BsBt −Bs−rBt+r = eipi(s−r)BrBt−s+r.BsBt −Bs−rBt+r= [−eipit(−βt)αs − eipis(−βs)αt + eipi(t+r)(−βt+r)αs−r+ eipi(s−r)(−βs−r)αt+r]/32= Br[eipi(s−r)(−βs−r)αs−r(αt−s+r − eipi(t−s+r)(−βt−s+r)]= eipi(s−r)BrBt−s+r.Again from (4), we get (17) as follows.Bs+t+1 = Bs+1Bt+1 −BsBt.Bs+1Bt+1 −BsBt= [αs+t+1(α− α−1) + eipi(s+t+1)(−β)s+t+1(β − e−ipi(−β)−1)]/32= [2√8αs+t+1 − 2√8eipi(s+t+1)(−β)s+t+1]/32= Bs+t+1.18Using (4) and (5), we get (18) as follows.16B2s = C2s ± eipis.[2√8Bs2√8Bs]/2 = [α2s + e2ipis(−β2s)− 2eipisαs(−βs)]/2= C2s − eipis(−1)s= C2s + (−1)s+1eipis.Using (4), we get (19) as follows.32B3s = B3s ± 3eipisBs.32B3s = [2√8Bs]3/2√8= [α3s − e3ipis(−β3s)− 3eipisαs(−βs)[αs − eipis(−βs)]/2√8= [2√8B3s − 3eipis(−1)s2√8Bs]2√8= B3s + (−1)s+13eipisBs.19References[1] A. Behera and G.K. Panda, “On the Square roots of Triangular numbers”, Fi-bonacci Quarterly, 37 No. 2, 1999, 98-105.[2] Berczes, A., K. Liptai, I. Pink, Fibonacci Quarterly, “On Generalized Balancingnumbers”, Vol. 48, 2010, No. 2, 121-128.[3] Piero Filipponi, “Real Fibonacci and Lucas numbers with Real subscript”, 1991.[4] Thomas Koshy, “Elementary and Number Theory And Applications”.[5] Thomas Koshy, “Fibonacci And Lucas number With Applications”.[6] G.K. Panda and P.K. Ray, “Co-Balancing numbers and Co-Balncers”, Interna-tionals Journals of Mathematics and Mathematical Sciences. Vol. 2005, No. 8,1189-1200.[7] G.K. Panda and S.S.Rout, “Gap Balancing Number”.[8] G.K. Panda and S.S. Rout, “A class of recurrent sequences exhibiting some ex-citing properties of Balancing numbers”, Int. J. Math. Comp. Sci., 6(2012), 4-6.[9] G. K. Panda, “Some fascinating properties of balancing numbers”, to appear inFibonacci Numbers and their Applications, Vol. 10, 2006.[10] P.K. Ray, “Balancing and cobalancing numbers”, Ph.D. Thesis, National Insti-tute of Technology, Rourkela, 2009.[11] P.K. Ray, “Balancing sequences of matrices with applications of Algebra of Bal-ancing numbers”, Vol. 20, 2014.[12] R. Witula, “Fibonacci and Lucas Numbers for real indices and some applica-tions”, ACTA PHYSICA POLONICA A, Vol. 120, 2011.20

Balancing and Lucas-balancing Numbers With Real Indices

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Balancing and Lucas-balancing Numbers With Real Indices

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