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A COMPLETE CHARACTERIZATION OF B-POWER FRACTIONS THAT CAN BE REPRESENTED AS SERIES OF GENERAL n-BONACCI NUMBERS JIN-ZA5 LEE and JIA-SHENG LEE Soochow University, Taipei, Taiwan, R.O.C. (Submitted April 1985) INTRODUCTION AND MAIN RESULT U In 1953, Fenton Stancliff [5] noted that -^r = .0112358 89 £ !0<k = k =0 13 21 + 1) Fk, where Fk denotes the kth Fibonacci number. Until recently, this expansion was regarded as an anomalous numerical curiosity, possibly related to the fact that 89 is a Fibonacci number (see Remark in [5]), but not generalizing to other fractions in an obvious manner. In 1980, C. F. Winans [6] showed that the sums £ 10"(/c+ 1)Fa^ approximate 1/71, 2/59, and 3/31 for a = 2, 3, and 4, respectively. Moreover, he showed that the sums £ l O ~ 2 ( k + 1 ) F a k approximate 1/9899, 1/9701, 2/9599, and 3/9301 for a = 1, 2, 3, and 4, respectively. Since then, several authors proved general theorems on fractions that can be represented as series Involving Fibonacci numbers and general n-Bonacci numbers [1, 2, 3, 4 ] . In the present paper we will prove a theorem which includes as special cases all the earlier results. We introduce some notation in order to state our theorem. Let arbitrary complex numbers AQ9 Al9 ..., Am, W0, Wl9 ..., Wm, and B be given. Construct the sequence W^ by the recursion m ^n + m+ 1 ~ L ArWn + m_ r for n ^ 0 or, equivalently, by the formula m r =0 for any integer n where a)r (r = 0, 1, ...,777) are the zeros of the polynomial q(z) = zm+l m - E r=0 and (A0, X1$ equations m E^r^r p=0 72 ..., Xm) ~ ^n Arzm-r is the unique solution of the system of m + 1 linear (n = 0 , 1, . . . , 77?) [Feb. S-POWER FRACTIONS AS SERIES OF GENERAL n-BONACCI NUMBERS (see [2], p, 35). Finally., we introduces for any integer a, M(m) = fl (B - < ) . p= o Theorem: For integers a > 1, 3 > 0, and any complex 5 satisfying max I ud^/B I < 1, we have the formula oo m M(m) • L Z T ^ = 5 • E Aro>r - + 6 ^ = 0 n (5-OJ"). 0 < ^ < 777 P = 0 Remark: In the above formula, M(m) and the right-hand side are in fact integers if B9 A Q 9 A 1 9 ..., A m , h\9 Wl9 ..., Wm are all integers.. Now we can comment on earlier results in more detail. In 1981, Hudson and Winans [1] handled the case of the ordinary Fibonacci sequence with 3 = 0? B = 10 n . According to [3] and [4], their result can be written as ±10-n(k + l)F , ? i02» *-i 9 n - 10 La - (-l)a where La denote the Lucas numbers. Also In 1981, Long [4] treated the case of the general Fibonacci sequence, i»es, m = 1 and arbitrary AQ9 A19 WQS W19 and Bs with the restrictions however, to a = 1, 3 = 0. In 1985, Kohler [2] gave the generalization for arbitrary ms A Q S A 1 9 . .., A m 9 WQ9 W1$ «»»s Wm$ B9 again with the restriction to a = I, g = 0. His result Is E B'kWk .= p(B)/q(B)9 where p is a polynomial of degree m with explicitly given coefficients. Also in 1985, Lee [3] discussed the cases m = 1 and m = 2 of general Fibonacci and Tribonacci sequences with arbitrary a and g. The results of [3] will be deduced from our Theorem in Examples 1 and 2 below., For this purpose, we introduce the notation 77? oo p= 0 Proof of Theorem: k =0 We have oo oo k=0 k=0 r=0 7/7 - „ r=0 \k-0 I r=0 Convergence is guaranteed by the condition on B. 1987] 5-0)" In the same ways we obtain 73 S-PQWER FRACTIONS AS SERIES OF GENERAL n-BONACCl NUMBERS Y,B'kwak+o = E U ' k=l r=Q Multiplying with M(m) yields the Theorem. Remark: Partial sums of the series in our Theorem can be expressed by these series, according to the formula B"- t B'kWak+6 = B». ± B-«Wak+e + E k=0 k =0 EXAMPLES 2. Example 1: The general Fibonacci sequence. Wn + 2 ~ A^Wn +1 B-*Wak+(m^y k = X Take m = 1. + A1Wn, M{1) = (B - <^)(B - UJ«) = B2 - BSa + t B~kwak+Z fc = i Then we have = U 0 < + S ( B - 0)«) + \^l+\B = (BWa+B - (-AJ*, - o.p}/M(l) (-A^W^/Md), £ B-kWak+B = B(B^ - (-VX-a)/W)- fc = 0 As to t h e p a r t i a l sums, we g e t £.Bn-\k = B"L(l)/M(l) - £ B-kWak + k=0 m+fi k=l BnL(l) -BWa(n+1) (-Al)aWm+e +&+ B2 - BSa + C-Axf These formulas a r e e q u a l t o Theorems 1-3 of [ 3 ] . Example 2: The g e n e r a l T r i b o n a c c i s e q u e n c e . Wn+3 = A0Wn +2 + A,Wn+1 M{2) = B 3 - B2Sa Take m = 2 . + A2Wn, + BAa2S_a - A\, t^Wa+z = (B%+& tQB-kW^B = B{B2Wz + B(Wa+ii - SaWB) + ZBn'kWak+& k=0 + B(W2a+& - SaWa+B) = BnL(2)/M(2) Then we o b t a i n B-kWak - t + + Aa2W^)/M(2), Aa2WB_a)M2), an+6 k=l (continued) 7k [Feb. 3-POWER FRACTIONS AS SERIES OF GENERAL n-BONACCI NUMBERS *^< 2 > ~ BX<n+i) + B+B(SaWa(n+1) B3 - B2Sa + &-Wa{n+2) + B)- Aa2Wm+i + BAa2S_a - Aa2 These formulas are equal to (9) and Theorems 7 and 8 in [3], and a misprint in Theorem 7 in [3] is corrected. Example 3: The general Tetranacci sequence. W n + , = ^ + 3 A W + + l n+2 M(3) = Bh - B3Sa + B2(S2 Take m = 3. A W 2 n +l + A 3Wn > - B(-A3)aS_a - S2a)/2 Then we have +M 3 ) \ fc = 1 + 5 ( - 4 3 ) a ( 5 _ a ^ - ^B_a) - oo ZB~kwak+Q k=o = B{B% + B2(Wa+$ - SaW$) + B(2A/ 2 a + 6 - 25 a A/ a+6 + (S2 " S2a)WB)/2 11 Bn Wak+3 - {BnL(3) - B3W0i(n+l) - B(~A3f (~A3)aWB}/M(3), +B B2(Wa(n+2) (S.aWan +6 (~A3)aW&_a}/M(3)5 + Q - SaWa(n+1) + Q) - Wa(n+ 1) + B ) + (-^ 3 ) a ^ + 6 }/M(3). Formulas for m ^ 4 can be obtained in a similar manner. ACKNOWLEDGMENTS We are deeply thankful to Dr. Hong-Jinh Chang and to the referee for the earnest discussions and valuable suggestions made. REFERENCES 1. 2. 3. 4. 5. 6. R. H. Hudson & C. F. Winans. "A Complete Characterization of the Decimal Fractions That Can Be Represented as E 10~k(t+ 1 V a ^ , Where Fai Is the aith Fibonacci Number." The Fibonacci Quarterly 19, no. 5 (1981):414-421. G. Kohler. "Generating Functions of Fibonacci-Like Sequences and Decimal Expansions of Some Fractions." The Fibonacci Quarterly 23, no. 1 (1985): 29-35. J. S. Lee. "A Complete Characterization of £-Power Fractions That Can Be Represented as Series of General Fibonacci Numbers or of General Tribonacci Numbers." Tamkang J. of Management Sciences 6, no. 1 (1985):41-52. C. T. Long. "The Decimal Expansion of 1/89 and Related Results." The Fibonacci Quarterly 19, no. 1 (1981):53-55. F. Stancliff. "A Curious Property of au ." Scripta Mathematica 19 (1953): 126. C. F. Winans. "The Fibonacci Series in the Decimal Equivalents of Fractions." A Collection of Manuscripts Related to the Fibonacci Sequence. Santa Clara, Calif.: The Fibonacci Association, 19809 pp. 78-81. •<>•<>• 1987] 75