REDUCTION FORMULAS FOR THE SUMMATION OF RECIPROCALS IN CERTAIN SECOND-ORDER RECURRING SEQUENCES

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1 REDUCTION FORMULAS FOR THE SUMMATION OF RECIPROCALS IN CERTAIN SECOND-ORDER RECURRING SEQUENCES R. S. Melham Department of Mathematical Sciences, University of Technology, Sydney PO Box 23, Broadway, NSW 2007, Australia (Submitted November 999). INTRODUCTION In [2], Brousseau considered sums of the form S(KK..;K) = ± FF 7X* * -,AJ = j ; where the k t are positive integers with k l <k 2 <--<k m. (.2) could be written as l F n=\ A n A n+k^ n+k 2 "' A n+k m F 0.) ( " i r ', (.2) n=\ r n r n+k\n+k 2 ''' n+k m He stated that the sums in (.) S(k h k 2,...,k m ) = r^r 2 S(\2,...,m) (.3) T(k h k 2,...,kJ = r 3 + r 4 T(l,2,...,in), (.4) where A\, r 2, r 3, r 4 are rational numbers that depend upon k x,k 2,...,k m. He arrived at this conclusion after treating several cases involving small values of m. Our aim in this paper is to prove Brousseau f s claim by providing reduction formulas that accomplish this task. Recently, Andre-Jeannin [] treated the case m = by giving explicit expressions for the coefficients r x, r 2, r 3, r 4. Indeed, he worked with a generalization of the Fibonacci sequence, we will do the same. In light of Andre-Jeannin's results, we consider only m>2. We have found, for each of the sums (.3) (.4), that two reduction formulas are needed for the case m- 2, three are needed for m> 3. Consequently, we treat those cases separately. Define the sequences {UJ {WJ for all integers n by \U = pu^-qu _ 2, U 0 = 0,U l = l, \W n = pw n _ x -qw n _ 2, W,=a,W x = b. Here a, b, p, q are assumed to be integers with pq*0 A = p 2-4q > 0. Consequently, we can write down closed expressions for U W n (see [3]): TT a n -p". m Aa n -Bj3",. where a = (p + J~A)/2, fi = (p-<ja)/2, A=h-afi, B = h-aa. Thus, {WJ generalizes {UJ which, in turn, generalizes {FJ. 2002] 7

2 We note that a>\ a n d a > ^ i f p > 0, w h i l e /? < - l \p\ > \a\ if p<0. Consequently, W *-^:a n a-p * ' Throughout the remainder of the paper, we take ifp>0,mdw n * ^~p n ifp<0. n a-fi. 8(k l,k 2,...,k m )-2^ww w w (.6) (.7) (.8) T{k l,k 2,...,kJ = ^ 00 n=l ^Ki+k^Ki+k 2 f ly - * ^Ki+k m (.9) where the k t are positive integers as described earlier. From (.6) it follows that U n & 0 for n >. We shall suppose that W n * 0 for n >. Then, by (.6) (.7), use of the ratio test shows that the series in (.8) (.9) are absolutely convergent. We require the following identities: UJV n l -W nhii = qu m _ l W H, (.0) Um-k+fflt+k ~ W n+m = qu m _ k W njtk _ x, (-) UJV n + q U d W n _ m _ d = U m+d W _ d, (.2) PW n+m +q 2 U m _ 2 W n = U m W n^ (.3) Um-l+Wn+k ~ Uk-l+ffli+m = *? ^m-k^th-l-l' (-4) Identity (.) follows from (.0), which is essentially (3.4) in [3], where the initial values of {[/ } are shifted. Identities (.3) (.4) follow from (.2), which occurs as (5.7) in [4]. 2, THREE TERMS IN THE DENOMINATOR Our results for the case in which the denominator consists of a product of three terms are contained in the following theorem. Theorem : Let k x k 2 be positive integers with k x <k 2. Then &{k ly k 2 ) qll, ki~k\ -[U.^Sikt -, k 2 ) - S(k, -, k,)] if < k,, (2.) v Uy T(k h k 2 ) = -^ [U^^Tik, q 2 u k u>. S(l,k 2 -l)- I WW. if 2 < A:, -, k 2 ) - T(k, -, *,)] if < K (2.2) (2.3) q 2 u* T{\,k 2 ) = ^-T{\,2)+ ^- 2 ww 2 w k. -T(l,k 2-l) if 2 < >t, (2.4) 72 [FEB.

3 Proof: With the use of (.), it follows that W n W n+k W n + ki WJY^W^ WJV^.JV^ summing both sides we.obtain (2.). Likewise, to obtain (2.3), we first multiply (2.5) by (-)"" sum both sides. Next we have (2.5) ^ P +.. ' lukr- " 2 ";, (2.6) " n+wn+k 2 WJ n+ J n+2 W n +JY n + 2 W n+ki which follows from (.3). Now, if we sum both sides of (2.6) note that ^ ^W n+ xw n+2 W v 2 n+k2 ' ' W x W 2 W ki > we obtain (2.2). Finally, to establish (2.4), we multiply (2.6) by (-l)"", sum both sides, note that This proves Theorem. D y H)"" = \ ~r(i 9 k 2 -i). n=l ^n+wn+'wn+k ^WWk 2 It is instructive to work through some examples. Taking W n -F n using (2.) (2.2) repeatedly, we find that S(3,6) = ^ + \S(\ 9 2), this agrees with the corresponding entry in Table III of [2]. Again, with W n = F, we have 7(3,6) = - ^ + ± 7 (, 2 ). 3* MORE THAN THREE TERMS IN THE DENOMINATOR Let k l9 k k m be positive integers put P{k h... 9 k m ) = WJV n+ki... W n+km. With this notation, the work that follows will be more succinct. The main result of this section is contained in the theorem that follows, where we give only the reduction formulas for S(k l9 k k m ). After the proof, we will indicate how the corresponding reduction formulas for T(k l9 k 2, -,k m ) can be obtained. Theorem 2: For m > 3, let k x < k 2 < < k m be positive integers set k 0 = 0. Then S(k l9 k k m ) = k.-ka\ TT S(k l kj mml9 kj-l 9 kj k m _ 2 9' c m) q J U k _ k U k k + j k m _ x -k~+ijj S(k h... 9 k J _ l9 k J.-l 9 k j k m _ l )? U k m -k m _ l ifl<j<m-2 kj_t <kj-l; (3.) S(k l9..., k m ) - m &>-> S(k l9..., k m _ l9 k m _ x, k m ) S(k l9..., k m _ 29 k m _ x, k m _ l ) it k m _ 2 < k m _ l ; (3.2) 2002] 73

4 S(l,2,...,m-l,kJ = ^S(l,2,...,m), ^K-m K m e use of (.4), we see that SYl m l /r > A(W~,*f \k m ) j ^ ^ ^ if m < k n (3.3) _f K m K m-\ U k m -k J+ i r{k ly..., kj-i, kj, kj,..., ^w_2, w ) >,-.+ P{k x,..., kj_ h kj l,kj,..., k m _ l ) summing both sides we obtain (3.). Next we have? /, Km km U km -k m _ x +\ P{k l,...,k m ) P\k\,.^k m _ 2,k m _ l \k m ) P{ki k m _ 29 k m _ l, k m _ ] ) which can be proved with the use of (.). Summing both sides, we obtain (3.2). Finally, with the aid of (.2) we see that Uk U <l m U k m K m _ ^ m i J K m m P{\,2,...,m-\,k m ) P(l,2,...,m) W n^n+ 2...W n^n+ K The reduction formula (3.3) follows if we sum both sides observe that ^ = 5(,2,...,m-\,k,, m,x -)- ^W W WW JW2...w m w, w=l rr n+l rr n+2 ' rr n+m rr n+k m This completes the proof of Theorem 2. As was the case in Theorem, the reduction formulas for Jean be obtained from those for S. In (3.) (3.2), we simply replace S by T. In (3.3), we first replace the term in square brackets b y -5(,2,...,m-,* M -) Wt-WJTk. then replace 5 by T. As an application of Theorem 2 we have, with W n - F, 5(,2,4,6,7) = -35(,2,3,4,7) + 25(,2,3,4,6) by (3.); 5(,2,3,4,7) = ^ + A 5 (, 2, 3, 4, 5 ) - ^ 5 (, 2, 3, 4, 6 ) by(3.3); 5(,2,3,4,6) = j^+5(,2,3,4,5) by (3.3). (3.4) (3.5) (3.6) Together (3.4)-(3.6) imply that 5a2 ' 4 ' 6 ' 7) = d 6-2>' 2 ' 3 ' 4 ' 5) - 74 [FEB.

5 4. CONCLUDING COMMENTS Recently, Rabinowltz [5] considered the finite sums associated with (.) (.2). That is, he took the upper limit of summation to be N, gave an algorithm for expressing the resulting sums in terms of Ij-. IV - ' Zyr- In addition, he posed a number of interesting open questions. REFERENCES. Richard Andre-Jeannin. "Summation of Reciprocals in Certain Second-Order Recurring Sequences." The Fibonacci Quarterly 35. (997): Brother Alfred Brousseau. "Summation of Infinite Fibonacci Series." The Fibonacci Quarterly 7,2 (969): A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 33 (965): Ray Melham. "Generalizations of Some Identities of Long." The Fibonacci Quarterly 37.2 (999): Stanley Rabinowitz. "Algorithmic Summation of Reciprocals of Products of Fibonacci Numbers." The Fibonacci Quarterly 372 (999): AMS Classification Numbers: B39, B37, 40A ] 75

(Submitted November 1999) 1. INTRODUCTION - F. n+k 2. n=1f" n+k l. <... < k m

(Submitted November 1999) 1. INTRODUCTION - F. n+k 2. n=1f n+k l. <... < k m REDUCTION FORMULAS FOR THE SUMMATION OF RECIPROCALS IN CERTAIN SECOND-ORDER RECURRING SEQUENCES R. S. Melham Department of Mathematical Sciences, University of Technology, Sydney PO Box 123, Broadway,