#A40 INTEGERS 13 (2013) FINITE SUMS THAT INVOLVE RECIPROCALS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS

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1 #A40 INTEGERS 3 (203) FINITE SUMS THAT INVOLVE RECIPROCALS OF PRODUCTS OF GENERALIZED FIBONACCI NUMBERS R S Melham School of Mathematical Sciences, University of Technology, Sydney, Australia raymelham@utseduau Received: 2/4/2, Accepted: 4/30/3, Published: 6/24/3 Abstract In this paper we find closed forms for certain finite sums In each case the denominator of the summand consists of products of generalized Fibonacci numbers Furthermore, we express each closed form in terms of rational numbers Introduction The Fibonacci and Lucas numbers are defined, respectively, for all integers n, by F n = F + F n 2, F 0 = 0, F =, L n = L + L n 2, L 0 = 2, L = Define, for all integers n, the sequences {U n } and {V n } by U n = pu + U n 2, U 0 = 0, U =, () V n = pv + V n 2, V 0 = 2, V = p, (2) in which p is a positive integer Then {U n } and {V n } are integer sequences that generalize the Fibonacci and Lucas numbers, respectively Throughout this paper p is taken to be a positive integer Let = p Then with the use of standard difference techniques it can be shown that the closed forms (the Binet forms) for U n and V n are U n = αn β n α β, V n = α n + β n, (3) where α = p + /2, and β = p /2 Next we define, for all integers n, the sequence {W n } by W n = pw + W n 2, W 0 = a, W = b, (4) I dedicate this paper to my mother Maria She continues to be a fountain of love and support

2 INTEGERS: 3 (203) 2 where a 0 and b 0 are integers with (a, b) = (0, 0) These conditions on a and b, together with the fact that p is assumed to be a positive integer, ensure that each of the reciprocal sums in this paper is well defined It can be shown that W n = Aαn Bβ n, (5) α β where A = b aβ and B = b aα Associated with {W n } is the constant e W = AB = b 2 pab a 2, which occurs in the sequel Accordingly, e F = and e L = 5 An identity linking the Fibonacci and Lucas numbers is L n = F + F n+, and a similar identity links the sequences {U n } and {V n } Motivated by this we define a companion sequence W n of {Wn } by W n = W + W n+ With the use of (5) we see that W n = Aα n + Bβ n (6) The sequences {W n } and W n generalize the sequences {Un } and {V n }, respectively Furthermore U n = V n, and V n = U n, so that F n = L n, and L n = 5F n Let k, m 0, and n 2 be integers In this paper (in Section 3) we give a closed expression, in terms of rational numbers, for each of the following sums: S (k, m, n) = T (k, m, n) = T 2 (k, m, n) = T 3 (k, m, n) = X (k, m, n) = ( ) ki W ki+m W k(i+)+m, (7) ( ) ki U k(i+)+m W ki+m W k(i+)+m W k(i+2)+m, (8) ( ) ki V k(i+)+m W ki+m W k(i+)+m W k(i+2)+m, (9) ( ) ki W k(i+)+m W ki+m W k(i+)+m W k(i+2)+m, (0) W ki+m W k(i+)+m W k(i+2)+m W k(i+3)+m () In Section 2 we present some background and motivation for our study In Section 3 we present our main results, and in Section 4 we present a detailed proof of one of our results The method of proof that we outline can be used to prove all the results in this paper It is reasonable to surmise that there are sums, analogous to (7)-(), with longer products in their denominators, for which closed forms can be found In Sections 5, 6, and 7 we select some of (7)-() and show that this is the case

3 INTEGERS: 3 (203) 3 2 Background and Motivation The broad question of summation of reciprocals that involve Fibonacci or generalized Fibonacci numbers has a long history In this short paper it is not our intention to give this history Instead, in the next few paragraphs, we give a brief commentary on work that we have cited on reciprocal sums that involve the summands in (7)-() After this, we indicate the motivation for the present paper André-Jeannin [] considered the summand in (7) for the particular cases W n = U n and W n = V n Taking m = 0 and k an odd integer, he expressed the infinite sums in terms of the Lambert Series L(x) = x i, x < xi Inspired by the work of André-Jeannin [], the authors in [5] considered the analogues of U n and V n for the recurrence W n = pw W n 2, and obtained analogues of André-Jeannin s results for these sequences The authors first obtained two finite sums in terms of the irrational roots of x 2 px + = 0 They then took the appropriate limits to obtain the corresponding infinite sums Interestingly, these infinite sums did not involve the Lambert Series, but were expressed in terms of the irrational roots of x 2 px + = 0 Filipponi [3] considered the summand in (7) for the particular cases W n = U n and W n = V n Taking m = 0 and k an even integer, he expressed the infinite sums in terms of α and β André-Jeannin [2] considered the more general sequence of integers defined by W n = pw qw n 2, in which the initial values W 0 and W are integers, and p and q are integers with pq = 0 For this sequence he studied the sums W ki+m W k(i+i0)+m and q ki+m, (2) W ki+m W k(i+i0)+m for integers k, m 0, and i 0 In the course of his analysis, André- Jeannin expressed any finite sums in terms of one of the roots of x 2 px + q = 0 Furthermore, in (2), he evaluated only the sum on the right in terms of rational numbers In [6] we studied the infinite sums W k(i+m) W ki W k(i+m) W k(i+2m), (3) and ( ) i W ki W k(i+m) W k(i+2m) W k(i+3m), (4)

4 INTEGERS: 3 (203) 4 in which k and m are odd integers In [7] we studied the infinite sums and W k(i+m) W ki W k(i+m) W k(i+2m), (5) W ki W k(i+m) W k(i+2m) W k(i+3m), (6) in which k and m are integers with k even For (3)-(6) it was only in the case of (5) that we managed to express the infinite sum as a finite sum of rational numbers Reflecting upon the results in the foregoing paragraphs, we realized that sums (finite or infinite) involving summands that are similar to those in (7)-() are not usually evaluated in terms of rational numbers It was this realization that prompted us to embark upon the investigation that led to the present paper We list two finite sums (see [8]) that we recently evaluated in terms of rational numbers F ki+m F k(i+)+m = where k > 0 is an even integer, and m > 0 is any integer L ki+m L k(i+)+m = F k() F k F k+m F kn+m, n >, (7) F k() F k L k+m L kn+m, n >, (8) where k = 0 is an even integer, and m is any integer In the present paper our evaluation of S (k, m, n) generalizes both (7) and (8) 3 The Main Results We now state our main results As stated in Section, in (9)-(23) k, m 0, and n 2 are assumed to be integers We have U k W k+m S (k, m, n) = ( )k U k() W kn+m, (9) e W U 2k W k+m T (k, m, n) = U k W k U k() W kn+m + ( )k+ W k U kn W k(n+)+m + ( )k+ W k W 2k+m, (20)

5 INTEGERS: 3 (203) 5 e W U 2k W k+m T 2 (k, m, n) = U k W k U k() W kn+m + ( )k+ W k U kn W k(n+)+m + ( )k W k W 2k+m, (2) U k W k+m T 3 (k, m, n) = ( )k ( )kn W k+m, (22) W 2k+m W kn+m W k(n+)+m e W V k U 3k W k+m X (k, m, n) = ( )m W 4k+m + ( )k+m Uk() W 2k+m W 3k+m Uk 2 W kn+m + ( )k+ V 2k U kn W k(n+)+m + U k(n+) W k(n+2)+m (23) To illustrate, we present two examples of (23) Let k = and m = 0 Then for W n = F n (23) becomes = 7 F i F i+ F i+2 F i+3 4 F + 3F n + F n+ (24) 2 F n F n+ F n+2 Let k = 2 and m = 0 Then for W n = L n (23) becomes = L 2i L 2(i+) L 2(i+2) L 2(i+3) 432 F2() 7F 2n + F 2(n+) 360 L 2n L 2(n+) L 2(n+2) (25) 4 The Method of Proof In this section, to illustrate our method of proof, we prove (23) in detail We require the following four identities, which can be proved with the use of the closed forms: and U kn W kn+m U k() W k(n+)+m = ( ) k(n+) U k W k+m, (26) U 4k W 2k+m ( ) k U k W 3k+m U 2 k W 4k+m = ( ) k U 3k W k+m, (27) W 2k+m W 3k+m W k+m W 4k+m = ( ) k+m e W U 2 k V k, (28) W kn+m W k(n+)+m V 2k W kn+m W k(n+3)+m (29) + W k(n+2)+m W k(n+3)+m = ( ) kn+m e W U k V k U 3k

6 INTEGERS: 3 (203) 6 For n 2 denote the right side of (23) by R (k, m, n) Then R (k, m, n + ) R (k, m, n) = ( )k+m U kn Uk 2 U k() W k(n+)+m W kn+m + ( )m+ V 2k Uk(n+) Uk 2 W k(n+2)+m + ( )k+m Uk(n+2) Uk 2 W k(n+3)+m With the use of (26) the right side becomes ( ) kn+m W k+m U k W kn+m W k(n+)+m and with the use of (29) we see that (30) simplifies to V 2k U kn W k(n+)+m U k(n+) W k(n+2)+m W k(n+)+m W k(n+2)+m + W k(n+2)+m W k(n+3)+m, (30) Thus Next, e W V k U 3k W k+m W kn+m W k(n+)+m W k(n+2)+m W k(n+3)+m (3) R (k, m, n + ) R (k, m, n) (32) = e W V k U 3k W k+m (X (k, m, n + ) X (k, m, n)) R (k, m, 2) = ( )k+m U 2 k Uk + ( )k+ U 4k + U 3k W 2k+m W 3k+m W 4k+m (33) + ( )m W 4k+m W 2k+m W 3k+m Expressing the right side of (33) as a fraction with denominator U 2 k W 2k+mW 3k+m W 4k+m, the numerator is ( ) m+ W 4k+m U4k W 2k+m ( ) k U k W 3k+m U 2 k W 4k+m + ( ) k+m U 3k W 2k+m W 3k+m Then, with the use of (27) and (28), we see that (34) R (k, m, 2) = ( )k+m U 3k (W 2k+m W 3k+m W k+m W 4k+m ) U 2 k W 2k+mW 3k+m W 4k+m (35) e W Uk 2 = V ku 3k Uk 2W 2k+mW 3k+m W 4k+m e W V k U 3k = W 2k+m W 3k+m W 4k+m = e W V k U 3k W k+m X (k, m, 2)

7 INTEGERS: 3 (203) 7 Taken together, (32) and (35) show that (23) is true We remark that all the results in this paper can be proved in a similar manner Above we made use of identities (26)-(29) to assist in the proof An alternative method is to simply use brute force Specifically, to prove that two quantities are equal we substitute the closed forms of the sequences in question and expand with the use of a computer algebra system All of the results in this paper can be proved with this method With this method any occurrence of e W is replaced by AB 5 Sums that Belong to the Same Family as T 3 (k, m, n) Throughout the remainder of this paper k, m 0, and n 2 are assumed to be integers Let us write (22) as U k W k+m W 2k+m T 3 (k, m, n) = ( ) k ( )kn W k+m W 2k+m W kn+m W k(n+)+m (36) Now define the following sum, where the denominator of the summand consists of seven factors T 7 (k, m, n) = ( ) ki W k(i+3)+m W ki+m W k(i+)+m W k(i+6)+m (37) In (37) we have modified our notation to make it more suggestive Specifically, the number 7 in the subscript of T 7 denotes seven factors in the denominator of the summand We continue this convention in what follows Next, define the following sum, where the denominator of the summand consists of eleven factors T (k, m, n) = ( ) ki W k(i+5)+m W ki+m W k(i+)+m W k(i+0)+m (38) Finally, define the following sum, where the denominator of the summand consists of fifteen factors T 5 (k, m, n) = ( ) ki W k(i+7)+m W ki+m W k(i+)+m W k(i+4)+m (39) In each of (37)-(39) the analogy with (0) is clear Furthermore, the pattern in (36) can be extended to yield U 3k W k+m W 6k+m T 7 (k, m, n) = ( ) k ( )kn W k+m W 6k+m W kn+m W k(n+5)+m, (40)

8 INTEGERS: 3 (203) 8 and U 5k W k+m W 0k+m T (k, m, n) = ( ) k ( )kn W k+m W 0k+m W kn+m W k(n+9)+m, (4) U 7k W k+m W 4k+m T 5 (k, m, n) = ( ) k ( )kn W k+m W 4k+m W kn+m W k(n+3)+m (42) The lists of formulas (37)-(39) and (40)-(42) have clearly defined patterns and can easily be extended by the reader Are there similar sums, for which closed forms exist, that have 5, 9, 3, factors in the denominator of the summand? We have discovered such sums The first few representatives of these sums are t 5 (k, m, n) = t 9 (k, m, n) = t 3 (k, m, n) = W k(i+2)+m W ki+m W k(i+)+m W k(i+4)+m, (43) W k(i+4)+m W ki+m W k(i+)+m W k(i+8)+m, (44) W k(i+6)+m W ki+m W k(i+)+m W k(i+2)+m (45) We have found that U 2k W k+m W 4k+m t 5 (k, m, n) = W k+m W 4k+m W kn+m W k(n+3)+m, (46) U 4k W k+m W 8k+m t 9 (k, m, n) = W k+m W 8k+m W kn+m W k(n+7)+m, (47) U 6k W k+m W 2k+m t 3 (k, m, n) = W k+m W 2k+m W kn+m W k(n+)+m (48) We have not been able to find closed forms for those counterparts to (43)-(45) where each summand is multiplied by ( ) ki The sums (43)-(45) together with their respective closed forms (46)-(48) have clearly defined patterns and can easily be extended by the reader Let k = 2 and m = 0 Then for W n = F n (40) becomes L 2(i+3) F 2i F 2(i+) F 2(i+6) = Let k = and m = 0 Then for W n = L n (40) becomes ( ) i F i+3 L i L i+ L i+6 = F 2n F 2(n+) F 2(n+5) (49) 6632( )n L n L n+ L n+5 (50)

9 INTEGERS: 3 (203) 9 6 Sums that Belong to the Same Family as S (k, m, n) Define the following sum, where the denominator of the summand consists of six factors ( ) ki S 6 (k, m, n) = (5) W ki+m W k(i+)+m W k(i+5)+m We have managed to find a closed form for (5) To present this closed form succinctly we define three quantities c i = c i (k) as follows: Then c 0 =, c = ( ) k+ V k V 3k, c 2 = ( )k V 6k + V 2k + 2( ) k 2 U k(n 2) i=0 e 2 W U k U 5k (S 6 (k, m, n) S 6 (k, m, 2)) = (52) 2 c i + W (2+i)k+m W (n+i)k+m W (6 i)k+m W (n+4 i)k+m Indeed, numerical evidence suggests that a similar closed form exists for the analogous sum S 0 (k, m, n) with ten factors in the denominator of the summand, and for the analogous sum S 4 (k, m, n) with fourteen factors in the denominator of the summand Numerical evidence also suggests that similar closed forms exist for analogous sums that have 8, 22, 26, factors in the denominator of the summand Let k = 2 and m = 0 Then for W n = F n (52) becomes F 2i F 2(i+) F 2(i+5) 44 = F 2(n 2) 3F 2n 27 4F 2(n+) F 2(n+2) 55F 2(n+3) + (53) 44F 2(n+4) Let k = and m = 0 Then for W n = L n (52) becomes ( ) i L i L i+ L i = F n 2 3L n + L n L n+3 + 8L n+4 7L n+2 (54)

10 INTEGERS: 3 (203) 0 7 Sums that Belong to the Same Family as X (k, m, n) Define the following sum, where the denominator of the summand consists of eight factors X 8 (k, m, n) = (55) W ki+m W k(i+)+m W k(i+7)+m We have managed to find a closed form for (55) In order to present this closed form we define four quantities f i = f i (k) as follows: Then f 0 =, f = U 3kV 4k U k, f 2 = ( )k U 3k V8k + ( ) k V 2k +, U k f 3 = ( )k+ V 4k V8k + V 4k + 2( ) k V 2k e 3 W U k U 7k (X 8 (k, m, n) X 8 (k, m, 2)) = (56) 3 ( ) m U k(n 2) f i + W (2+i)k+m W (n+i)k+m W (8 i)k+m W (n+6 i)k+m i=0 Numerical evidence suggests that similar closed forms exist for analogous sums that have 2, 6, 20, factors in the denominator of the summand Let k = and m = 0 Then for W n = F n (56) becomes 320 F i F i+ F i+7 2 = F n 2 F n 7 F n+ 30 F n F n+4 4 3F n+5 + 2F n+6 F n+3 (57) 8 Concluding Comments We have discovered closed forms for variants of (8)-(0) that we do not present here For instance, we have discovered a closed form for ( ) ki U ki+m W ki+m W k(i+)+m W k(i+2)+m (58) Furthermore, in the spirit of Sections 5-7, we have discovered lengthier analogues of these variants The possibilities seem endless

11 INTEGERS: 3 (203) Acknowledgment of priority: Just prior to submitting this article I was made aware of Theorem in [4] In fact our result (9) is a consequence of Theorem in [4], as is our main result in [8] I thank Zhi-Wei Sun for alerting me to [4] References [] R André-Jeannin, Lambert series and the summation of reciprocals in certain Fibonacci- Lucas-type sequences, Fibonacci Quart 28 (990), [2] R André-Jeannin, Summation of reciprocals in certain second-order recurring sequences, Fibonacci Quart 35 (997), [3] P Filipponi, A note on two theorems of Melham and Shannon, Fibonacci Quart 36 (998), [4] Hong Hu, Zhi-Wei Sun & Jian-Xin Liu, Reciprocal sums of second-order recurrent sequences, Fibonacci Quart 39 (200), [5] R S Melham & A G Shannon, On reciprocal sums of Chebyshev related sequences, Fibonacci Quart 33 (995), [6] R S Melham, Summation of reciprocals which involve products of terms from generalized Fibonacci sequences, Fibonacci Quart 38 (2000), [7] R S Melham, Summation of reciprocals which involve products of terms from generalized Fibonacci sequences-part II, Fibonacci Quart 39 (200), [8] R S Melham, On finite sums of Good and Shar that involve reciprocals of Fibonacci numbers, Integers 2 (202), #A6