SOME FIBONACCI AND LUCAS IDENTITIES L. CARLITZ Dyke Uiversity, Durham, North Carolia ad H. H. FERNS Victoria, B. C, Caada 1. I the usual otatio, put (1.1) F _<**-{? = -=- -, L = a a + /3 a - p ^ where (1.2) a = (1 +V5), p = 4(1 - V5) f * 2 = or + 1, 2 = + 1, ap = -1 It is rather obvious that polyomial idetities ca be used i cojuctio with (1.1) ad (1.2) to produce Fiboacci ad Lucas idetities. For example, by the biomial theorem* 2 a <» " - 1 : ("V s which gives tt-3) F 2 = E ( s ) F s L 2 = E (, s /) L s * ad ideed Supported i part by NSF Grat GP 5174. 61
62 SOME FIBONACCI AND LUCAS IDENTITIES [Feb. ( 1 4 ) F 2+k = 2 ( s ) F s + k s L 2+k = S ( s ) L s - &» X 7 x where k is a arbitrary iteger. Agai, sice a 3 = a(a + 1) = 2a + 1, we get i the same way <"> -3,* - E (:)>*.*. ^+k = E 0 ^ More geerally, sice it follows that a T = F a + F - 9 r r - 1 <«> *»* E (") *K3u- v* - E (")»» r t s=o x s=o This ca be carried further. For example, sice (1.7) a 2m = L m «m - < - l ) m, we get (1.8) *2m+k Z-# v L) s m ms+k \ / T = - V" /( 1x(-s)(m+l)/\ ^(-sjtm+d/v T s T ij 2m+k " ZJI { ~ l) s V m s + k
1970] SOME FIBONACCI AND LUCAS IDENTITIES 63 The idetity (1.7) geeralizes to (1.9) a =! l 2 m f f - (_l) m F - 1 > m m m Ideed (1.9) is equivalet to which is obviously true. From (1.9) s we obtai (1.10) F F = V / 1)( -s)(m.+l)^-s F s m rm+k xlr (r-l)m rm ms+k 9 7 T y ^ / ^(-smm+l^-s ^ s m rm+k x J v ~ ; (r-l)m rm ms+k With each of the above idetities is associated a umber of related idetities. as either For example, we may rewrite r a = F a + F - r r - 1 r r a - JP a = F * or a - F ^ = F a Hece we obtai r r - 1 r - 1 r (1.11) E ( - i,s (y F F = F F JLX r r(-s)+s+k r - 1 k J2 (-D ( s ) F? L r ( - s ) + s + k = F r - 1 v S=0 L k
64 SOME FIBONACCI AND LUCAS IDENTITIES [Feb. ad (1.12) W)»-sQ F : s F = F F rs+k r +k rs+k r +k We remark that (1.9) ca be geeralized eve further, amely to (1.13) F a sm - F a = < - l ) s m F,,, rm sm (r~s)m ad (1.10) ca ow be exteded i a obvious way. 2. Additioal idetities are obtaied by makig use of formulas such as (2.1) a 2 + 1 = av ~. Note that (2.2) js 2 + 1 = -j3v5. Thus we get so that E (sv 2s = 5/2 * S (sv s = ( - 1) 5/2 ^ ^ S / \ /o Ta+k. i x ^+k ^ s j ^ s + k ~ a - 8 / \ F = 5 /2 a - (-1) P It follows that
1970] SOME FIBONACCI AND LUCAS IDENTITIES 65 (2.3) 2s+k 5 ( - l ) / 2 L +k ( eve) ( odd), Similarly (2.4) Eft)- 2s+k 5 /2 L +k ( eve) 5 < + 1 ) / 2 F + k ( odd) We omit the v a r i a t s of (2 8 3) ad (2,4). We ca geeralize (2 e l) a s follows^ (2.5) 2m 2 m - 1 2m+l x T /=- «= L 9 +1 2m+l a " L 9. ^A 2m The correspodig formulas for a r e (2.6) 02m = -F Q p\/s - L 0, 2m 2 m - l ^ - - W.«>* We therefore get the followig geeralizatios of (2.3) ad (2.4): (2.8) Z2 W L 2m-1 2ms+k 5 F 2m +k 1 2m +k ( ( eve) odd), (2.9) X ) (s) L 2m-l J 2ms+k ( 5 / 2 F L, ] 2m +ic (m+l)/2 ( 5 F 2m F +k ( ( eve) odd), (2.10)!>> M ( ) L 2 m + 1 F 2m(-s)++k 0 ( eve) - ( D - D / 2 ^ 2 5 V F 2 m ( o d d ) >
66 SOME FIBONACCI AND LUCAS IDENTITIES i /2 2 / (2.11) E ^ - Q L ^ L ^ ^ ^ 2m 0 ( We omit the variats of (2.8),, (2.11). 3, I the ext place, O the other had, a + «*) a + >*-E(;WS(:)= x)». v C V r x r v c v r=0 N / 2 k = yx k v( i V r - k=0 L«d r=0 ~i \rf^l - r / (1 + a x r ( l + x) 11 = (1 + a 2 x + ax 2 ) E r=0 E r=0 2 E k=0 / \ r r, 1 la x (a 4 x) r w Q«r * r E s=g 3s<2k ( ^ r " S X ) ( k 3 s X s\ 2kr -3s Comparig coefficiets of x we get <«> E &.*" > - E ( k!,)( k ; > 2k - 3 " r=0 v 3s-2k v x
1970] SOME FIBONACCI AND LUCAS IDENTITIES 67 It therefore follows that *» S(;)(^r)ry- E L 1 s)( k.- > 2k. 3s+) r=0 W X 3s 2k X X ad r=0 3s 2k L 2k-3s+j for all j. I particulars for k = s these formulas reduce to E *«, E (.".)(*) F 2-3s+j r=0 3s ^2 ad ««E(;)! Vj= E ( )(?>, 2-3s+j, r=0 * 3s^2 respectively,, We have similarly a + A,» (1 -*,» = (») ^ w> s (S> r=0 s 2 k k=0 T=Q f
68 SOME FIBONACCI AND LUCAS IDENTITIES (1 + Qf2 x ) (l - x) = (1 + ax - a*x 2 f r=0 x r r /- xr x (1 - ax) r =0 ^ / x = E «k - k E w ) t " s ) ( k ; s ) k=0 2s<k V I X Comparig coefficiets of x we get * > E <-«^(;)( k r)«2r - «k E <-» B ( k s)( k -. *) r=0 It follows that x / x 2s^k M k E w> k - r (;V k - - W, -»*, E <-«t - s)( k r=0 W X f 2s<k X X k «.«s < W ; ) ( k» V r + J - L k+j 2 <-» s ( k!.f r=0 x x / I particular, for k =, we get 2s k \ / X»» E W u V ^, - * +i E <-» s (t) ft) r=0 ^ 2s< (sao, <-!,»-(;)\ r+). v. <-W a " s )M r=0 x 2s^ \ I \ I
1970] SOME FIBONACCI AND LUCAS IDENTITIES 69 More geeral results ca be obtaied by usig (1.7). shall omit the statemet of the formulas i questio, For brevity, we 4. The formulas k=o V / k=o \ / x were proposed as a problem i this Quarterly (Vol. 4 (1966), p. 332, H-97). The formulas k were also proposed as a problem (Vol. 5 (1967), p* 70, H-106). They ca be proved rapidly by makig use of kow formulas for the Legedre polyomial. We recall that [ l, 162, 166] (4.5) ->-±M-->)M k=0 s0! (-) k (-r=s-"iww
70 SOME FIBONACCI AND LUCAS IDENTITIES If we take u = (x + l ) / ( x - 1), we get (4.6) (4.7) k=0 x ; k=0 \ / > / t(^» k -E<- u ""t)(^k> k,»-»"" k - k=o ^ / k=o > N Multiplyig both sides of (4.6) by ir ad the take u = a,p. a - 1 = a, we get zq! * kti -zq( ; k )" i+k - k=0 X k=0 > / x / It follows that (4.8) k=0 w k=0 x / x / (4.9) Z ^ t k ) L k+j = L U \ k ) L j+k- k=0 % f k=0 F o r j = 0, (4.8) ad (4.9) reduce to (4.1) ad (4.2), respectively. If i the ext place we replace u = a 2 9j3 2 i (4.6), we get t&^-utiy k+j
1970] SOME FIBONACCI AND LUCAS IDENTITIES 71, % 0 ^ (:)( : ^ k+j so that (4.10) Z-#tkl F 2k+j Z ^ l k / I k J F -k+j k=0 x 7 k=0 X I X I (4.11) Z ^ t k ) L 2k+j " Z ^ ( k ) ( k ) L -J k=0 X / k=0 % f X These formulas evidetly iclude (4.3) ad (4.4). I exactly the same way (4.7) yields k=0 X / k=0 \ / \ I F 2k+j- z@\^i>"i;)(\ + j L 2k+j- 9 k=0 x k=0 ad k=0 X / k=0 W V / k=0 X / k=0 X f X f
72 SOME FIBONACCI AND LUCAS IDENTITIES [Feb. The idetities (4.8),, (4.15) ca be geeralized further by employig, i place of (4.5), the followig formulas for Jacobi polyomials 1, 255 : e U) «=Z(-i)( r)(^) k (^) " k /A + V^/V + ^ + 1) k/x-l\ k \ )h\v (X + 1) * v 2 / -MIH^^-r where (X + l ) = (X + 1)(X + 2)--- (X + ), (X + l)o = 1 The fial results are - sc-x-sm-ise^ 1, k F j+k- + 1 ) i L j-tk- k=0 X X / X k=0 \ / <> E(,; x 11 X-"K)-(^jE^) ( + X + M- + D k (A + D t F -k+j <- E(^X->- Ms ^^^w
1 9 7 0 ] SOME FIBONACCI AND LUCAS IDENTITIES 73 «-> ze^(:-^-(^m^^-- ) F k j+2k-* k=0 k=0 **> Z(T)(^H +i (>: ) (;) ^ * w Z(\ +A )( :^2k+1 - (>:%(i) ^ P w k=0 v k=0 ( + A +fi + 1). We remark that takig u = -or i (4.6) leads to (4.24, ± l-w") F k+j - ± <-l> -*(») (» J ") r 2. 2k+., k=0 w =0 w \ / (4.25, ( - «* (» ) L +.., -!, - * (» ) (» k * ) L 2. 2 k +., k=0 k=0 ad so o, Some of the formulas i the earlier part of the paper are certaily ot ew. However, we have ot attempted the rather hopeless task of fidig where they first occurred. I ay evet, it may be of iterest to derive them by the methods of the preset paper. It would, of course, be possible to fid may additioal idetities. REFERENCE 1. E. D. Raiville, Special Fuctios, Macmilla, New York, 1960. * *