CERTAIN GENERAL BINOMIAL-FIBONACCI SUMS J. W. LAYMAN Virgiia Polytechic Istitute State Uiversity, Blacksburg, Virgiia Numerous writers appear to have bee fasciated by the may iterestig summatio idetitites ivolvig the Fiboacci related Lucas umbers. Various types of formulas are discussed various methods are used. Some ivolve biomial coefficiets [2], [4]. Geeratig fuctio methods are used i [2] [5] higher powers appear i [6]. Combiatios of these or other approaches appear i [1], [3] [7]. Oe of the most tatalizig displays of such formulas was the followig group of biomial-fiboacci idetities give by Hoggatt [5]. He gives: < 1 > l U F 2 = ( l ) F k, (2) 2 F 2 = E {l) F 3k, (3) 3 F 2 = { k)f 4 k. k=o I these formulas throughout this paper F deotes the/7 (4) F = F._ 1+ F ^2, F 0 = 0, F t = 1. Fiboacci umber defied by the recurrece: Hoggatt attributes formula (2) to D. A. Lid, (3) to a special case of Problem 3-88 i the Fiboacci Quarterly states that (1) is well kow. The three idetities give above suggest, rather strogly, the possibility of a geeral formula of which those give are special istaces. Hoggatt does obtai may ew sums but does ot appear to have succeeded i obtaiig a satisfactory geeralizatio of formulas (1)-(3). I the preset paper, we give elemetary, yet rather powerful, methods which yield may geeral biomial- Fiboacci summatio idetities. I particular, we obtai a sequece of sums the three simplest members of which are precisely the formulas (1)-(3) give above. I additio, similar families of sums are obtaied with the closed forms a^ mf a 3 mf^ for/77 = /, 2, 3,, as well as the geeral two-parameter family of sums with the closed form (a r)m ) F r. Our pricipal tools for obtaiig sums will be the biomial expasio formula m (5) Z ( )(y-d k = y m, the fact that the Fiboacci umber F is a liear combiatio of a b, where a=l±jl b = l^fare roots of the polyomial equatio (6) x 2 =x+1. The Fiboacci umbers are the (7) F = (1/^~5)(a -b ). 362
DEC. 1977 CERTAIN GENERAL BINOMIAL FIBONACCI SUSVIS 363 We are already i a positio to obtai a summatio formula. Let w st for a root a orb of (6). The we have (8) w 2 = w+ 1. Clearly, the, by (5) (8), therefore u=0 =0 (a 2 ) -(b 2 ) = (I) (a k -b k ). =0 But from (7), this is see to be equivalet to f 2 = E (l) F k> =0 which is formula (1). I order to obtai more geeral results, we proceed as follows. From (8) we see that, i geeral, by a easy iductio, w = w+1 = F2W + F 1, w = w + w = Fjw + F2, (9) w m = w ' 1 +w m ~ 2 = F m w+f m^ Rewritig, we have or, equivaletly,,, m F 1- -2 = - -p^w, m t 1, r m~l r m~l -te)""" = (' ) (^r) (10) - I - ^ - 1 w" = V ( " ) I -=L.) k w mk where, agai, w may be either a or b. Agai usig the fact that F is a liear combiatio of a obtai b, we (^)" f '- (^-""1^)' f - -" Equatio (11) takes o especially simple forms for certai values of m. For example, whe m =2 J, respectively, we have (12) F = f ( k)(-v +k F 2k (13) 2 F = ( k)(-d k+ F 3k. Other values of m result i o-itegral ratios i (11), e.g.,/77 =4 5 give (14) ( 3 jyf H = (-ir ( h k)h'/*) k F 4k
364 CERTAIN GENERAL BINOMIAL FIBONACCI SUMS [DEC. (15) U) F = (-1) E { h)(-m h F 5 k, x ' k=o Each of the sums (12) (15) the geeral sum i (11) yield closed forms of the type ( a l,m) F, I order to obtai sums with closed forms of the type (a 2) m) F2 we retur to (9). If we let m = 2 solve for w, w = w - 1, we may substitute this expressio ito (9) to obtai (16) w m = F m (w 2 ~ 1) + F m i = F m w 2 -(F m -F m, 1 ) = F m w 2 - F m _ 2. This is equivalet to: (17) ~ ~ w 2 = - w m + l, m t 2. *~m~~2 'm~2 Now proceedig i the same maer as led to (11) results i the geeral formula (18) (F m ) F 2 = (I) (F m. 2 r k F mk, mi 2. k=o The special cases m = 1, 3, 4 of this geeral equatio are foud to give exactly the three sums ivolvig F 2 which were listed by Hoggatt give above i (1)-(3). All other cases ca easily be see to lead to formulas cotaiig a power of a Fiboacci umber i the summ i this sese previous ivestigators ca be said to have foud all "easy" sums of this type. The first two cases givig ew sums are thus, for/?? = 5 6, (19) S F 2 = f ) ( k)2-' k F 5k (20) 8 F 2 = ( k)3 ' k F 6kf Steps similar to those leadig to (16) ca be followed to express w m which, followig our geeral procedure, yields 2w m = F m w 3 + F m _ 3 i terms of w 3. We fid, after simplifyig, (21) (F m ) F 3 = ^ (l)(-ir k 2 k (F m 2 r k F mk, mi 3. k=i For/77 = 2, 4, 5, 6 we have, respectively, (22) F 3 = (-ir Z (l)(-2) k F 2k, (23) 3 F 3 = (-1J E (l)(-2) k F 4k, k (24) s F 3 = f-1) E {l)(-2) k F 5k,
1977] CERTAIN GENERAL BINOMIAL -FIBONACCI SUSVIS 365 (25) 4 F 3 = (-IP i( k)(-v k F 6k. k=o x Rather tha cotiuig with these special families of sums, we ow proceed to the geeral two-parameter family yieldig closed forms of the type Let 0 < r < m. From (9) we have which give, after cosiderable simplificatio, fa ) F i^^ra / r r - w r = F r w+f r i, w m = FmW+F^i (26) w m = F -f- w' + t-v 1 " 1 % - r, 0 < r < m. ' r ' r The result just obtaied is equivalet to.. / F \ (27) hirl /^V r = <-i) T [-P-\w m + i. \ r 'm-r J \ r 'm-r / which yields the summatio (28) (F m ) F m = Y,(l)(- 1 > r( ~ k) ( F r-rr k (F r ) k F m k, X ' 0 < r < m, valid for all itegral m,, /-satisfyig 0 < r < m. A umber of special cases of the above geeral formula have bee give previously i this paper for r= 1, 2, 3. Aother iterestig case results whe m = 2r. Usig the well kow fact that F2 r /F r \$ the Lucas umber Z. r defied by the recurrece (29) L = L ^+L, 2, L 0 = 2, L t = 1, we have, i this case, (30) (L r ) F r = [l)(-v r ( ^F 2rk, The special case r = 2p has bee obtaied by Hoggatt i [5]. Some istaces of (30) which have ot bee give amog our previous formulas are (3D 7 F 4 = E [l) F Bk k=o x ; (32) 11 F 5 = Z[ k)(-v ~ k F 10 k which obtai whe r= 4 5, respectively. Of course, if we recall that the Lucas umbers L are liear combiatios of a b, defied i (7), specifically (33) L = a +b, the we see that each sum obtaied above remais valid whe L is substituted for F at the appropriate occurreces of F i each formula. We state some of these. From (18) we have
366 CERTAIN GENERAL BINOMIAL FIBONACCI SUMS DEC. 1977 (34) (F m 2 = E ( l)(fm-2) ~ k L mk, mt2, K ' several specific istaces beig ^L2 = S ( k) L k> V ' 2 " %L 2 = X [ k) L 3k X ' 3?l[ -2 = J2 \ k) L 4k. V ' The iterested reader may obtai other Lucas umber aalogs of formulas give above. Prelimiary results idicate that modificatios of the methods used i this paper will lead to may other quite geeral results o biomial Fiboacci sums. Perhaps we might be forgive for paraphrasig Professor Moriarty (see [4]) i sayig "may beautiful results have bee obtaied, may yet remai." REFERENCES 1. L. Carlitz, Problem B-135, The Fiboacci Quarterly, 2. C. A. Church M. Bickell, "Expoetial Geeratig Fuctios for Fiboacci Idetities," The Fiboacci Quarterly, Vol. 11, No. 3 (Oct. 1973), pp. 275-281. 3. H. Freitag, "O Summatios Expasios of Fiboacci Numbers," The Fiboacci Quarterly, Vol. 11, No. 1 (Feb. 1973), pp. 63-71. 4. H. W. Gould, "The Case of the Strage Biomial Idetities of Professor Moriarty," The Fiboacci Quarterly, Vol. 10, No. 4 (Dec. 1972), pp. 381-391. 5. V. E. Hoggatt, Jr., "Some Special Fiboacci Lucas Geeratig Fuctios," The Fiboacci Quarterly, Vol.9, No. 2 (April 1971), pp. 121-133. 6. V. E. Hoggatt, Jr., M. Bickelil, "Fourth Power Fiboacci Idetities from Pascal's Triagle," The Fiboacci Quarterly, Vol. 2, No. 3 (Oct 1964), pp. 261-266. 7. D. Zeitli,"O Idetities for Fiboacci Numbers," Amer. Math. Mothly, 70 (1963), pp. 987-991. *******