A POWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES

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1 A OWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES V. E. Hoggatt,., San ose State College, San ose, Calif. and D. A. Lind, Univesity of Viginia, Chalottesville, V a. 1. INTRODUCTION The following hold fo all integes n and k: F n+k ~ F k F n+i + F k-l F n F n + k = k, ) F ^ + ( F k F k- 2 ) F n + 1 " ( W k - ^ ' F n + k = ( F k F k-i F k- 2 / 2 > F n < F k F k-i F k-3) F n +2 " ( F k F k- 2 F k- 3 ) F n + i " < F k-i F k- 2 F k- 3 / 2 K These identities suggest that thee is a geneal expansion of the fom (1.1) jjy F n + k = E a 3=o ik)f U Hee we show such an expansion does indeed exist, find an expession fo the coefficients a.(k s p), and genealize (1.1) to second ode ecuent sequences. 2 S A FIBONACCI OWER IDENTITY Define the Fibonomial coefficients ml :\ by F F F m m-i m-+i ( > 0); F 1 F F = 1 aden [4] poved that the tem-by-tem poduct z of p sequences each obeying the Fibonacci ecuence satisfies 2 7 4

2 [Oct. 1966] A OWER IDENTITY FOR 275 SECOND-ORDER RECURRENT SEQUENCES p+i (2.D 2 H-> 3 ( + I ) / 2 fo integal n. shown that the deteminant p + 1 In p a t i c u l a z n = F ^ obeys (2.1). Calitz, [ 1, Section 1] has n-j D "n++s (, s = 0, l, -, p ) has the value D p = ( - l ) ( + 1 ) ( n + 1 ) / 2 ( ). ( F ^. - F p ) ^ 0 i=o \ / implying that the p + 1 sequences {F },{F },, {F } a e linealy independent ove the e a l s. Since each of these sequences obeys the (p + 1) o d e e c u e n c e elation (2.1), they must span the space of solutions of (2.1). Theefoe an expansion of the fom (1.1) exists. To evaluate the coefficients a.(k,p) in (1.1) we fist put k 0,l s, p, giving a.(k,p) = 6. 7 fo 0 < j? k < p, whee 6.-. is the Konecke-delta d e - fined by <5., = 0 if j 4= k, 6-,, = 1. Next we show that the sequence obeys (2.1) fo j = 0, 1,, p. { a. ( k, p ) } l ^ o Indeed fom (1.1) we find p+i o = 2 <-D (+1)/2 + 1 "n+k- p I p+i j=o j =o + 1 a ( k -, p ) } F; n+j But the F v. (j = 0, 1, e * e,p) a e linealy independent, so that n+j E =o <- 1 > (+1)/2 p + 1 a.(k -, p ) = 0 ( j = 0, l, - - -, p )

3 276 A OWER IDENTITY FOR [Oct. Now conside b.(k,p) = (F, F, F, )/F,.(F.F. " «F F > 0 0 F. ) fo j = 0,!,,p - 1, b (k,p) = I pj, togethe with the convention that F 0 /F 0 = 1. Clealy bj(k,p) = <5., fo 0 < j,k < p. Since {b.(k,p)}j_ is ;the tem-by - tem poduct of p Fibonacci sequences, it must obey (2.1). Thus {a.(k,p)}, oo th ^ and b.(k,p)}, obey the same (p + 1) ode ecuence elation and have thei fist p + 1 values equal (j = 0, l, - -,, p ), so that a.(k,p) = b.(k,p). Since F_ = (-l) n+1 F, it follows that (~)(-+3)/2 F - F. = F. F 1 (-l) -1 j- p p- 1V ' so that fo j = 0,1,,p - 1, we have a.(k,p) = (-1) (lh)(-+3 )/2 I VW" F1 ll( F - F i)(v]"-^fl F k-j/ (-i) (-)(-+*)/2 ( ClDf] F / k - F k - j > ' which is also valid fo j = p using the convention F 0 / F 0 = 1. Then (1.1) becomes k (2 v(p-3)(p-+3)/21 ' 2) F n+k = 2 ^ ( " i L. < F k ^ / F k - j > ^ fo all k We emak that since consecutive p powes of the natual numbes obey 5 ( " " i ( - 1 ) ( n + ) = ' 3=o a development simila to the above leads to (2.3) (n + k)*- = 2 ^ e l ) <-' (:)(;)(-) j=o (n + if

4 1966] SECOND-ORDER RECURRENT SEQUENCES 277 a esult paallel to (2.2) 3. EXTENSION TO SECOND-ORDER RECURRENT SEQUENCES We now genealize the esult of Section 2. linea ecuence elation Conside the second-ode (3.1) y n + 2 = y n q y n (q * 0). Let a and b be the oots of the auxiliay polynomial x 2 - px + q of (3.1), Let w be any sequence satisfying (3,1)? and define u by u = ( a n - b n ) / (a - b) if a + b, and u = na if a = b s so that u also satisfies (3.1). n fml Following [ 4 ], we define the u-genealized binomial coefficients I I by u u * u,, i [ T ml = _m_m-i n^+i Tml = j u UiU 2 ---u _ 0 j u aden [4] has shown that the poduct x of p sequences each obeying (3.1) satisfies the (p + 1) ode ecuence elation p+i (3.2) ^TVl)^" 1 )/ 2 =o p + 1 ^ - 3 = If all of these sequences ae w, then it follows that x = w p obeys (3.2). It is ou aim to give the coesponding genealization of (1.1) fo the sequence w ; that is, to show thee exists coefficients a.(k,p,u) = a.(k) such that le (3 ' 3) ^ + k = I ] a 3 ( k ) w n+j j=o and to give an explicit fom fo the a.(k) e Calitz [ 1, Section 3 ] poved that V W ) = K++sl < ' s = O.L.)

5 278 A OWER IDENTITY FOR [Oct. is nonzeo, showing that the p + 1 sequences ae linealy independent Reasoning as befoe, we see these sequences span the space of solutions of (3.2), so that the expansion (3.3) indeed exists. utting k = 0,l,-**,p in (3.3) gives a.(k) = 6., fo 0 < j, k < p. It also follows as befoe that the sequence satisfies (3.2). Now conside b.(k.p.u) = b.(k) = u, u, / u, /u,-.(u.u. / u 1 u_ 1 *-*u. ) k k-i k - p ' k-f j j - i I 1 j ~ p ' fo j = 0, l,.. -, p - l, b p (k) = [^, along with the convention u 0 /u 0 = 1 Then bj(k) = 6.-. fo 0 < j, k < p. Also{b-.(k)}, obeys (3.2) because it is the poduct of p sequences each of which obeys (3.1). Since {a.(k)v? th 3 ^~ and {b-(k)}? (j = 0,l,»,p) obey the same (p + 1) ode ecuence elation and agee in the fist p + 1 values, we have a.(k) = b.(k). Now ab = a, so that u^ = (a" n - b ~ n ) / ( a - b) = -q n u. Then ~l j - p p-j n ; and thus fo j = 0,1,,p - 1 we see (3.4) a (k) = (.i)- q (-)(- +1 )/2 / U, U, U,, V / U U / U \ / U, \ / k k-i k-p+i W pp-i *. ]/ k-p\ \ Vp-l' " U l ' V ( u j- * * u i>< u p-j-'' u l> /1 u k-j / = (-l) " j q ( p " j ) ( p ~ 3 + l ) / 2. u u. (u, v /u,.), k-p k - j ' ' which is also valid fo j = p using the convention u 0 /u 0 = 1. Theefoe (3.3) becomes < 3-5 > p.-ki w n + k = Z ( - 1 ) " ',<p-i)(p-+1 )/*- (u, /u,.) w,. L u u v k-p/ k - j ' n+j

6 ] SECOND-ORDER RECURRENT SEQUENCES 279 Let Calitz has communicated and poved a futhe extension of this esult. () n n+aj n+a 2 n+a whee the a. ae abitay but fixed nonnega,tive integes. Then we have (3.6) 0» - n + k ^ E ^ 3=o - q (-)(- + 1 )/2 (u. 1 k-p /u. k-j.)x ; 9 n+j, whee u 0 /u 0 still applies. We note that putting a A = a 2 = e = a = 0 e - duces (3.6) to (3.5). To pove (3.6) using pevious techniques equies us to show that the sequences {?'} {«;}..{"&} ae linealy independent. To avoid this, we establish (3.6) by induction on k. Now (3.6) is tue fo k = 0 and all n. Assume it is tue fo some k ^ 0 and all n s and eplace n by n + 1, giving x ( p ) xi+k+i 5 > 1 > : u j=o p X> 1) ~ j+lq U=1 It follows fom (3.2) that -j q (-)(-+0/2 -+U-+I)(- +2 )/ 2 k-p X() n+j+1 V j ^ X() k- i+i u,( ) n+p+i K() n+p+i p+i p + 1 "S ( " 1)jq3(j " 1)/2 ^o -L(-)(- +1 )/ 2 j X() n+p+i-j u p + 1 n+j 3 -»u

7 280 Thus A OWER IDENTITY FOR [Oct x () n+k-h 1 k - -*U j=0 1 )-j q (-j)("j+l)/ ) 3 k-j+i 1 p+i k-j+i H k-p 3 Since V i V j + i - q " 3+lu k-p u j = V i V j + i ' we have ) n+k+i - u, k+il k + 1 i Z ( - 1 ) -L(-j)(-+i)/2 " 3 q u j=o E. i=0 ( 1) ~ q (~)(-+ 1 )/ 2-1 u. VitL x (p) u.u,.,, n+j 3 k-j+i u. k-p+i (p) V j + i n + j? completing the induction step and the poof e 4. SECIAL CASES In this section we educe (3.5) to a geneal Fibonacci powe Identity and to an identity involving powes of t e m s of an aithmetic pogession. F i s t if we let w = F,, u = F, whee and s a e fixed integes fo with n n s + ' n n s ' s 4 0, then both w and u satisfy (4.1) v - L y + (-1) v = 0. n+2' s n+i The oots of the auxiliay polynomial of (4.1) a e distinct fo s =f= 0 9 so that w and u satisfy the conditions of the pevious section,, In this c a s e the u - genealized binomial coefficients become the s-genealized Fibonacci F m l L u coefficients, defined by

8 1966] SECOND-ORDER RECURRENT SEQUENCES F F ins (m-i)s 1 (m-t+i)s F F e F (t > 0) ; = 1 ts ts-s s 281 A ecuence elation fo these coefficients is given in [3], Now hee q = <-l) S, SO (. D - q ^ M - ^ / 2 «e i ) ( ) [ s ( ^ l H ] / 2 Then (3.5) yields (4.3) F (n+k)s+ al)(-)[s(p-j-m)+2]/2 1=0 p.j. F (k-p)s F (k-)s F p < n + «s + utting s = 1 and = 0 gives equation (2,2). On the othe hand, if we let w = ns + and u - n, whee and s ' n n ' ae fixed integes, then w and u obey (4.4) V n+2-2v n-h + V n = 0 Since the chaacteistic polynomial of (4.4) has the double oot x = 1, both w and u satisfy the conditions fo the validity of (3.5). In this case we have q = 1 and, j =, 1, the usual binomial coefficient. Then (3.5) becomes (4.5) ([n + k]s + ) p f \p/\j/\k - j. (O + j]s + f This educes to (2.3) by setting s = 1 and = 0 S REFERENCES 1. L 9 Calitz, "Some Deteminants Containing owes of Fibonacci Numbes s?f Fibonacci Quately, 4(1966), No, 2, pp V. E 0 Hoggatt, a and A 0 Hillman, "The Chaacteistic olynomial of the Genealized Shift Matix/ 1 Fibonacci Quately, 3(1965), No 2 5 pp

9 282 A OWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES [Oct. 1966] 3. oblem H-72, oposed by V. E. Hoggatt,. Fibonacci Quately, 3(1965), No. 4, pp , 4. D. aden, "Recuing Sequences, n Riveon Lematimatika, eusalem (Isael), 1958, pp Roseanna F. Toetto and. Allen Fuchs,? Genealized Binomial Coefficients," F i b o n a c c i 2(1964), No e 4, pp ACKNOWLEDGEMENT The second-named autho was suppoted in pat by the Undegaduate Reseach aticipation ogam at the Univesity of Santa Claa though NSF Gant GY-273, * * REQUEST The Fibonacci Bibliogaphical Reseach Cente desies that any eade finding a Fibonacci efeence send a cad giving the efeence and a bief desciption of the contents. lease fowad all such infomation to; Fibonacci Bibliogaphical Reseach Cente, Mathematics Depatment, San ose State College, San ose, Califonia. * * * The Fibonacci Association invites Educational Institutions to apply fo Academic Membeship in the Association. The minimum subsciption fee is $25 annually. (Academic Membes will eceive two copies of each issue and will have thei names listed in the ounal. * * * * *