Reciprocal Sums of Elements Satisfying Second Order Linear Recurrences

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1 Chamchuri Journal of Mathematics Volume 00 Number, Reciprocal Sums of Elements Satisfying Second Order Linear Recurrences Kantaphon Kuhapatanakul and Vichian Laohakosol Received 30 Apr 00 Revised 6 Aug 00 Accepted 6 Aug 00 Abstract: Let {U n } n0 and {W n } n0 be two sequences defined by U 0 0,U, U n+ pu n+ + qu n and W n+ pw n+ + qw n W 0,W arbitrary with p,q Rp + 4q > 0 The aim of this paper is to prove N n q at U at n t W at nw at n+ at N+ at n W 0 W W q at p W at+n W at+n Wat W at W at N+ W at N+, where a, t N and t This identity generalizes a number of known identities such as n0 7 5 n, where { n } is the ibonacci sequence Keywords: reciprocal sum, linear recurrence, ibonacci numbers 000 Mathematics Subject Classification: B37, B39 Introduction Let { n } n 0 {0,,, n+ n + n,} Supported by the Commission on Higher Education and the Thailand Research und RTA and by Kasetsart University Research and Development Institute Corresponding author

2 94 Chamchuri J Math 00, no : K Kuhapatanakul and V Laohakosol and {L n } n 0 {,,,L n+ L n + L n,} denote the sequences of ibonacci, respectively, Lucas numbers In 974, Millin [9] posed the problem of showing that n0 n 7 5 A proof of by Good is given in [], while in [4], Hoggatt and Bicknell demonstrated eleven different methods of finding the same sum The identity was further extended by Hoggatt and Bicknell in [5], where they showed that 5a + L a a 5 if a is odd a n a n+ L a + a 5 otherwise a In [7], this last sum was also found to equal a+ + 5, a n a n+ while a finite version of this sum was shown by Greig [3] to be + n a an a even a a n i0 a i + a + an a odd a a a n In 976, using Lambert series expansions, Bruckman and Good [] evaluated several reciprocal sums including L k 3 n n0 k 3 n+ k 3 n L n0 k 3 n+ 5 k k k 3 5 k k 5 L k 4 Since the ibonacci and the Lucas numbers are elements satisfying the same second order linear recurrence relation but with different initial values, it is natural to ask whether the above-mentioned identities continue to hold for elements satisfying a general second order linear recurrence relation We answer this question affirmatively here

3 Reciprocal sums of elements satisfying second order linear recurrences 95 and Define three second order linear recurrences {U n } n 0, {V n } n 0 and {W n } n 0 by U n+ pu n+ + qu n, U 0 0, U V n+ pv n+ + qv n, V 0, V p W n+ pw n+ + qw n, W 0,W arbitrary, where p,q R are subject to p + 4q > 0 Let α,β be the two roots of its characteristic equation x px q 0 with β < α It is well-known, [6], that U n αn β n α β where r W W0β α β, V n α n + β n, W n r α n + r β n, and r W0α W α β If p,q, then U n n and V n L n are the ibonacci and Lucas numbers, respectively The following identities are easily verified n n L n 5 α n α n + n or β n β n + n 6 5α n αl n + L n or 5β n βl n L n 7 Certain reciprocal sums of elements in the sequence {U n } have previously appeared such as in 995, Melham and Shanon [8] found, when q, that U k U k + α if p > U n0 k n U k U k + β if p < 8 Results The notation of Section will be kept standard throughout Observe that the ibonacci and Lucas numbers satisfy the following identities In 9, taking m n +, we get m n m n n m n 9 L m L n L m L n n 5 m n, 0 n n+ n n, which is the result found by Cassini Theorem 53 in Chapter 5 of [6] These identities are special cases of in the next lemma

4 96 Chamchuri J Math 00, no : K Kuhapatanakul and V Laohakosol Lemma If m, n are two positive integers with m n, then W m W n W m W n q n W 0 W W U m n Proof Both sides of are zero when m n Observe from the recurrence relation that W m W m W n W n p q W m 0 W m q p q 0 0 p W n W n W m W m 3 p q p q p 0 0 }{{} n terms Evaluating the determinants on both sides, we get W n W n 3 q 0 W m n+ W W m n W 0 W m W n W m W n q n W m n+ W 0 W m n W It remains to show that W m n+ W 0 W m n W W 0 W W U m n, ie W k+ W 0 W k W W 0 W W U k, where m n+k Clearly, holds for k Assume it is true for an arbitrary positive integer i By definition and the induction hypothesis, we get W i+ W 0 W i+ W pw i+ + qw i W 0 pw i + qw i W pw i+ W 0 W i W + qw i W 0 W i W pw 0 W W U i + qw 0 W W U i W 0 W W U i+, ie, also holds for i + We next state and prove our main theorem

5 Reciprocal sums of elements satisfying second order linear recurrences 97 Theorem Let N,a,t N with t If W 0 W W and W n 0 for all n, then N q at p i W at+i W at+i Wat W 0W W W at Wat+N W at+n N q at U at i t i W Wat at iw at i+ W 0W W W at 3 N n U at n t W at nw at at n+ n N+ at W at N+ W at N+ p W at+n W at+n Proof Rewrite the equation of Lemma as q n U m n W n W m W 0 W W Putting m at + i,n at + i, we get q at p W at+i W at+i W 0 W W Wn W n W m 3 W m Wat+i 3 W at+i W at+i W at+i Summing over i from to N, we get the result of Part or Part, putting m at i+,n at i, we get q at U at i t W at iw at i+ W 0 W W Wati W at i The identity in Part follows by summing over i Part 3 follows by taking N atn+ at in Part W at i+ W at i+ 3 Applications Theorem is a host of a good deal of identities as we now show Corollary 3 If N,a,t N with t, then N q at p i U at+i U at+i Uat+N U at+n Uat N i q at p V at+i V at+i p +4q Vat V at U at 3 N q at U at i t i U U at N+ at iu at i+ U Uat at N+ 4 N q at U at i t i V Vat at iv at i+ p +4q V at Vat+N V at+n U at V at N+ V at N+

6 98 Chamchuri J Math 00, no : K Kuhapatanakul and V Laohakosol Proof or Parts and 3, put W n U n in Theorem Part and, respectively or Parts and 4, put W n V n in Theorem Part and, respectively Letting N in Corrolary 3, and using we obtain: U N V N lim lim N U N N V N α, Corollary 3 If a,t N with t, then i q at p U at+i U at+i α Uat U at q at p i V at+i V at+i Vat p +4q V at α 3 q at U at i t i U at iu at i+ α Uat U at 4 q at U at i t i V Vat at iv at i+ p +4q V at α Specializing certain parameters in Corrolary 3, several identities, mentioned in Section, follow easily as we illustrate now Putting t and q in Part 3, we get U i a i+ U a U a α which, after a little more computation, is 8 When {U n } { n } and {V n } {L n } are the ibonacci and Lucas sequences, we have α + 5/ Part 3 of Corrolary 3 gives, when t, the identity i a i+ a a + 5 a+ a a + 5, a a+

7 Reciprocal sums of elements satisfying second order linear recurrences 99 which is, and when t 3 and q, the identity i a3 i a3 i a3 i+ L a3 i i a3 i+ a 5 a 5 a 5 a a + 5 3a 3a + a a 3a + a a 3a a a 3a using 5 + a a 3a a 3a a a 3a using 9 a + a a a 3a 5 a a a a a 3a, using 6 which is 3 Part 4 of Corrolary 3 gives, when t 3, the identity i a3 i L a3 il a3 i+ a3 i L i a3 i+ a L 3a L 3a a 5 L a L 3a L 3a L a 5L a L 3a a 5 L a L 3a L 3a L a a 5 a 5L a L 3a using 5 using 0 5 a a 5L a a L a L 3a, using 7 which is 4 References [] PS Bruckman and IJ Good, A generalization of series of De Morgan, with applications of ibonacci type, ibonacci Quart, 4976, [] IJ Good, A reciprocal series of ibonacci numbers, ibonacci Quart, 4974, 346 [3] WE Greig, Sum of ibonacci reciprocals, ibonacci Quart, 5977, 46 48

8 00 Chamchuri J Math 00, no : K Kuhapatanakul and V Laohakosol [4] VE Hoggart, Jr, and M Bicknell, A primer for the ibonacci Numbers, part XV: variations on summing a series of reciprocals of ibonacci Numbers, ibonacci Quart, 43976, 7 76 [5] VE Hoggart, Jr, and M Bicknell, A reciprocal series of ibonacci Numbers with subscripts n k, ibonacci Quart, 45976, [6] T Koshy, ibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 00 [7] V Laohakosol and K Kuhapatanakul, The reverse irrationality criteria of Brun and Badea, Contributions in General Algebra II, Proc Intern Conf Discrete Math Appl, Bangkok, March 5-7, 008 a special volume published by East-West J Math, 7 34 [8] RS Melham and AG Shannon, On reciprocal sums of Chebyshev related sequences, ibonacci Quart, 33995, 94 0 [9] DA Millin, Problem H-37, ibonacci Quart, 3974, 309 Kantaphon Kuhapatanakul Department of Mathematics Kasetsart University Bangkok 0900, Thailand fscikpkk@kuacth Vichian Laohakosol Department of Mathematics Kasetsart University Bangkok 0900, Thailand fscivil@kuacth