1. INTRODUCTION. Fn 2 = F j F j+1 (1.1)

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1 CERTAIN CLASSES OF FINITE SUMS THAT INVOLVE GENERALIZED FIBONACCI AND LUCAS NUMBERS The beautiful identity R.S. Melham Deartment of Mathematical Sciences, University of Technology, Sydney PO Box 23, Broadway, NSW 2007 Australia (Submitted July 200 Final Revision January 2002). INTRODUCTION j Fn 2 = F j F j+ (.) n= was the insiration for [2], in which analogous sums involving cubes of Fibonacci numbers were develoed. In turn, [2] was the motivation for [5], [6], [7]. In the resent aer, where we restrict ourselves to summs that consist of roducts of at most two terms (as in (.)), our motivation has again been to find sums where the right side has a leasing form. We have found it rofitable to consider non-alternating sums, alternating sums, sums that alternate according to 2. Fibonacci-related sums of the latter tye are almost nonexistent in the literature. During the discovery rocess we became aware of numerous connections that exist between various grous of sums. So, rather than merely to resent a collection of sums, a riority of ours has been to highlight the strong thread of unity that exists. Another riority has been to achieve a balance between elegance generality,, to achieve this, exerimentation has led us to emloy the four sequences that we now define. We define the sequence {W n }, for all integers n, by W n = W n + W n 2, W 0 = a, W = b, (.2) where a, b, are assumed to be arbitrary comlex numbers with ( 2 + 2)( 2 + 4) 0. Then, since = , the roots α β of x 2 x = 0 are distinct. Hence the Binet form (see [3]) for W n is W n = Aαn Bβ n, (.3) α β where A = b aβ B = b aα. The Binet form gives W n for all integers n. We define another sequence {X n } by X n = W n+ + W n,, with the use of (.3), we find that X n = Aα n + Bβ n for all integers n. (.4) For (a, b) = (0, ) write W n = U n X n = V n. Thus W n X n generalize U n V n, resectively, which in turn generalize F n L n, resectively. Asects of {W n } {X n } have been treated, for examle, in [], [4], [], more recently in [8]. In Sections 2, 3, 4 we resent our results, then conclude by giving a samle roof. Since we have decided uon a limited focus, we do not claim that our results are exhaustive. Indeed, we exect that there is scoe for further research along the lines that we set forth. 47

2 In each of our sums the lower limit is allowed to vary. Accordingly, in keeing with convention, we always assume the uer limit to be greater than the lower limit, that either limit may be negative. 2. THE FIRST SET OF SUMS In this section, the sections that follow, we systematically consider non-alternating sums, alternating sums, sums that alternate according to 2. We begin with the following three results. { j W n = V (j i+)/2(w (j+i+)/2 + W (j+i )/2 ) if j i (mod 4) U (j i+)/2(x (j+i+)/2 + X (j+i )/2 ) if j i 3(mod 4), (2.) j ( ) n W n = ( ) j V (j i+)/2 (W (j+i+)/2 W (j+i )/2 ) if j i (mod 4) ( ) j U (j i+)/2 (X (j+i+)/2 X (j+i )/2 ) if j i 3(mod 4), (2.2) j 2 W n = { ( ) j(j+) 2 W (j+i)/2 (U (j i+2)/2 + ( ) j U (j i)/2 ) if j i 0(mod 2) ( ) j(j+) 2 U (j i+)/2 (W (j+i+)/2 + ( ) j W (j+i )/2 ) if j i (mod 2) (2.3) Now suose that, in the summs of (2.) (2.2), W is relaced by X. Then if j i 3 (mod 4) we modify the right sides by multilying with relacing each occurrence of X by W. It is now clear what (2.)-(2.3) become when any of U n, V n, or X n is substituted for W n in the summ. 3. THE SECOND SET OF SUMS In this section we consider sums similar to (2.)-(2.3) in which W n is relaced by W 2n.. { j W 2n = V j i+w j+i if j i 0(mod 2) U j i+x j+i if j i (mod 2), (3.) j X 2n = U j i+w j+i if j i (mod 2). (3.2) In the alternating sum that follows the arity of the limits is not an issue. j ( ) n W 2n = ( ) j U j i+ W j+i. (3.3) 48

3 For the following grou of sums the uer lower limits are secified as belonging to certain residue classes modulo 4. 4j + 2 W 2n = 2 U 4j 4iW 4j+4i+, (3.4) 4j +3 2 W 2n = 2 V 4j 4i 2X 4j+4i+3, (3.5) 2 W 2n = 2 U 4j 4i+4X 4j+4i+3, (3.6) +2 2 W 2n = 2 V 4j 4i+2W 4j+4i+5. (3.7) We now make an observation concerning the subscrits on the right sides of (3.)-(3.7). The subscrits of U V are (uer limit-lower limit+), while the subscrits of W X are (uer limit + lower limit). The reader can observe that this also alies to most of the sums in this aer. Notice that, in (3.4)-(3.7), one limit of summation is even while the other is odd. Accordingly, we have observed that each of (3.4)-(3.7) has a dual sum that is obtained with the use of the rule below. We highlight this rule since it also alies to certain grous of sums in Section 4. Rule for the Formation of the Dual Sum Relace the even limit by the even limit corresonding to the other residue class modulo 4 the odd limit by the odd limit corresonding to the other residue class modulo 4 Calculate the subscrits on the right in accordance with the aragrah following (3.7) Multily the right side by. For examle, the dual of (3.7) is 4j+ 2 W 2n = 2 V 4j 4i+2W 4j+4i+. (3.8) We also remark that if, in (3.5) (3.6), W in the summ is relaced by X, then on the right we relace X by W multily by. This also alies to the duals of (3.5) (3.6). It is now clear what the summation identities of this section become when W in the summ is relaced by either U, V, or X. 49

4 Finally if, in each sum of this seciton, W 2n is relaced by W 2n+k, the subscrit of W or X on the right side is simly increased by k. The same alies to each sum where X 2n occurs in the summ. 4. THE THIRD SET OF SUMS In this section we consider sums where the summ contains a second order roduct. Since, for examle, U n V n = U 2n, sums that involve the roduct U n V n follow from Section 3. However, sums that involve Un 2 or Vn 2 do not follow from anything we have done so far. To remedy this, to achieve more generality, we next consider sums in which the summ contains U n W n, U n X n, V n W n, or V n X n. In the three sums that follow i j are assumed to have different arities. Under this assumtion we have j U n W n = U j i+w j+i, (4.) j V n W n = U j i+x j+i, (4.2) j V n X n = U j i+w j+i, (4.3) It is clear that the corresonding sum involving the summ U n X n can be obtained from (4.). Next we consider the associated alternating sums, where, unfortunately, the right sides do not factorise nicely. Nevertheless, for the sake of comleteness, we have managed to salvage something in the sirit of our revious results. Without any assumtions on the arities of the limits we have j ( ) n U n W n = ( ( ) j ) U j i+ X j+i (j i + )X 0, (4.4) j ( ) n U n X n = ( ) j U j i+ W j+i (j i + )W 0. (4.5) In (4.4), if U n W n is relaced with V n X n, the right side is multilied by the sign of the coefficient of X 0 is changed. In (4.5), if U n X n is relaced by V n W n, then only the sign of the coefficient of W 0 is changed on the right side. The next grou of sums has 2 in the summ. 4j + 2 U n X n = 2 U 4j 4iW 4j+4i+, (4.6) 50

5 4j +3 2 U n X n = 2 V 4j 4i 2X 4j+4i+3, (4.7) 2 U n X n = 2 U 4j 4i+4X 4j+4i+3, (4.8) +2 2 U n X n = 2 V 4j 4i+2W 4j+4i+5 + 2W 0. (4.9) The right sides of these four sums should be comared, resectively, with the right sides of (3.4)-(3.7). The only difference occurs in (4.9) where the term 2W 0 is added. Furthermore, each of (4.6)-(4.9) has a dual sum that is obtained with the use of the rule in Section 3. And from this total of eight sums we obtain a further eight sums if, in each summ, we relace U n X n by V n W n. In each case the right side remains unchanged, excet for (4.9) its dual, where the coefficient of W 0 undergoes a change in sign. To conclude the list of sums in this aer we list four more,, as in the revious aragrah, we describe how to obtain an additional twelve. We have found the following. 4j + 2 U n W n = 2 U 4j 4iX 4j+4i+, (4.0) 4j +3 2 U n W n = 2 V 4j 4i 2W 4j+4i+3, (4.) +2 2 U n W n = 2 U 4j 4i+4W 4j+4i+3, (4.2) 2 U n W n = ( 2 V 4j 4i+2X 4j+4i+5 + 2X 0 ). (4.3) Each of (4.0)-(4.3) has a dual sum that is obtained with the use of the rule in Section 3. And from this total of eight sums we obtain a further eight sums if, in each summ, we relace U n W n by V n X n multily the right side by. Multilication by is the only change we are required to make to the right side, excet for (4.3) its dual, where, in addition, we must change the sign of the coefficient of X 0. For examle, the dual of (4.3) is 4j+ 2 U n W n = ( ) 2 V 4j 4i+2X 4j+4i+ + 2X 0, (4.4) 5

6 an additional sum is 4j+ ( 2 V n X n = 2 V 4j 4i+2X 4j+4i+ 2X 0 ). (4.5) By way of examle, if we ut W n = F n then (3.3), (3.5), (4.2) become, resectively, j ( ) n F 2n = ( ) j F j i+ F j+, (4.6) 4j +3 2 F 2n = 3 L 4j 4i 2L 4j+4i+3, (4.7) 2 F 2 n = 3 F 4j 4i+4F 4j+4i+3. (4.8) 5. A SAMPLE PROOF Each of our sums can be roved by induction, we illustrate by roving (4.2). We require the following result, which is a secial case of (79) in []. X n+k X n k = U k W n, k even. (5.) By our earlier assumtion on the limits, the smallest allowable value of j is i. Thus, we begin our inductive roof by verifying that 4i+3 2 U n W n = U 4W 8i+3. (5.2) We recall that = = (α β) 2, relace U n W n with their Binet forms, ex to obtain LHS = ((X 8i+6 X 8i+4 ) (X 8i+2 X 8i )) = (X 8i+5 X 8i+ ) (from the recurrence for X n ) = (X 8i+3+2 X 8i+3 2 ) = 2 W 8i+3 (from (5.)) 52

7 which is the right side of (5.2) since U 4 = Next, if (4.2) is true for the arameter j, then 4(j+)+3 2 U n W n = U 4j 4i+4W 4j+4i+3 + 4j+7 n=4j+4 we are required to rove that the right side of (5.3) is equal to Equivalently, we are required to rove that 4j+7 n=4j+4 Proceeding as before, we find 2 U n W n, (5.3) 2 +2 U 4j 4i+8W 4j+4i+7. 2 U n W n = (U 4j 4i+8W 4j+4i+7 U 4j 4i+4 W 4j+4i+3 ). LHS = (X 8j+4 X 8j+2 (X 8j+0 X 8j+8 )) = (X 8j+3 X 8j+9 ) RHS = the roof is comlete. = 2 W 8j+ (from (5.)). Similarly ( 2 + 2) (X 8j+5 X 8j+7 ) = ( 2 + 2) U 4W 8j+ 6. CONCLUDING COMMENTS = LHS, In [9] Russell considers the sums j R n j R ns n with no restrictions on the limits of summation. Here {R n } {S n } are sequences generated by the recurrence W n = W n + qw n 2 with q real. In each of these sums several cases are given deending on the values of q. Indeed, there are three cases for the first sum seven cases for the second. The character of Russell s sums ours is quite different, since our motivation has been to resent sums in which the right side has a leasing form. In a subsequent aer, Russell [0] gives finite sums in which each summ consists of roducts of u to three terms, where each term is generated by W n = W n + W n 2. Neither of Russell s aers contains alternating sums or sums that alternate according to 2. However, they are the only aers we have seen that contain finite sums where each summ consists of terms (or roducts of terms) generated by second-order linear recurrences, where the lower limit of summation is allowed to vary. We discovered each of our results numerically by first considering the Fibonacci Lucas sequences. Initially all lower limits were taken to be one, in each case we varied the uer limit until the sum could be exressed as a roduct of Fibonacci /or Lucas numbers. We then decided to vary the uer lower limits simultaneously then found that our 53

8 results could be translated to the more general sequences defined herein. The rocess was quite tedious, we gratefully acknowledge our use of the comuter algebra ackage Mathematica 3.0. REFERENCES [] G.E. Bergum & V.E. Hoggatt, Jr. Sums Products for Recurring Sequences. The Fibonacci Quarterly 3.2 (975): [2] S. Clary & P. Hemenway. On Sums of Cubes of Fibonacci Numbers. Alications of Fibonacci Numbers, 5: Ed. G.E. Bergum et al. Dordrecht: Kluwer, 993. [3] A.F. Horadam. Basic Proerties of a Certain Generalized Sequence of Numbers. The Fibonacci Quarterly 3.3 (965): [4] C.T. Long. Some Binomial Fibonacci Identities. Alications of Fibonacci Numbers, 3: Ed. G.E. Bergum et al. Dordrecht: Kluwer, 990. [5] R.S. Melham. Sums of Certain Products of Fibonacci Lucas Numbers. The Fibonacci Quarterly 37.3 (999): [6] R.S. Melham. Sums of Certain Products of Fibonacci Lucas Numbers-Part II. The Fibonacci Quarterly 38. (2000): 3-7. [7] R.S. Melham. Alternating Sums of Fourth Powers of Fibonacci Lucas Numbers. The Fibonacci Quarterly 38.3 (2000): [8] R.S. Melham. Summation of Recirocals which Involve Products of Terms from Generalized Fibonacci Sequences. The Fibonacci Quarterly 38.4 (2000): [9] D.L. Russell. Summation of Second-Order Recurrence Terms Their Squares. The Fibonacci Quarterly 9.4 (98): [0] D.L. Russell. Notes on Sums of Products of Generalized Fibonacci Numbers. The Fibonacci Quarterly 20.2 (982): 4-7. [] C.R. Wall. Some Congruences Involving Generalized Fibonacci Numbers. The Fibonacci Quarterly 7. (979): AMS Classification Numbers: B39, B37 54