Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences

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1 Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL , USA e-mal: Septembe 1, 00 1 Intoducton DeMove (1718 used the geneatng functon (found by employng the ecuence fo the Fbonacc sequence F x x 1 x x, to obtan the denttes F n αn β n, L n α n + β n (Lucas numbes wth α 1+, β 1. These denttes ae called Bnet fomulas, n hono of Bnet who n fact edscoveed them moe than one hunded yeas late, n 1843 (see [6]. Recpocally, usng the Bnet fomulas, we can fnd the geneatng functon easly F x 1 (α β x 1 ( 1 1 αx 1 x 1 βx 1 x x, snce αβ 1, α + β 1. A natual queston s whethe we can fnd a closed fom fo the geneatng functon fo powes of Fbonacc numbes, o bette yet, fo powes of any second-ode ecuence sequences. Caltz [1] and Rodan [4] wee unable to fnd the closed fom fo the geneatng functons F (, x of F n, but found a ecuence elaton among them, namely (1 L x + ( 1 x F (, x 1 + x ( 1 j A j j F ( j, ( 1j x, Also assocated wth the Insttute of Mathematcs of Romanan Academy, Buchaest, Romana [ ] j1 1

2 wth A j havng a complcated stuctue (see also []. We ae able to complete the study stated by them by fndng a closed fom fo the geneatng functon fo powes of any non-degeneate second-ode ecuence sequence. We would le to pont out, that ths fogotten technque we employ can be used to attac successfully othe sums o sees nvolvng any second-ode ecuence sequence. We also fnd closed foms fo non-weghted patal sums fo non-degeneate second-ode ecuence sequences, genealzng a theoem of Hoadam [3] and also weghted (by the bnomal coeffcents patal sums fo such sequences. Usng these esults we ndcate how to obtan some conguences modulo powes of fo expessons nvolvng Fbonacc and/o Lucas numbes. Geneatng Functons We consde the geneal non-degeneate second-ode ecuence, U n+1 au n + bu n 1, a, b, U 0, U 1 nteges, δ a + 4b 0. We ntend to fnd the geneatng functon of powes of ts tems, U(, x U x. It s nown that the Bnet fomula fo the sequence U n s U n Aα n Bβ n, whee α 1 (a + a + 4b, β 1 (a a + 4b and A U 1 U 0 β α β, B U 1 U 0 α α β. We assocate the sequence V n α n + β n, whch satsfes the same ecuence, wth the ntal condtons V 0, V 1 a. Theoem 1. We have U(, x f odd, and ( A ( AB B + ( b (B α A β x 1 ( b V x b x, 1 1 ( B U(, x ( AB + A ( b (B α + A β x 1 ( b V x + b x ( ( AB +, f even. 1 ( b x

3 Poof. We evaluate U(, x ( ( (Aα ( Bβ x ( A ( B (α β x ( A ( B 1 1 α β x. If odd, then assocatng, we get U(, x ( ( A ( 1 B 1 α β x A B 1 α β x 1 ( A ( 1 B A B + (A B α β A B α β x 1 (α β + α β x + α β x 1 ( A ( 1 B A B + ( b (A B α A B β x 1 ( b V x b x. 1 If even, then assocatng, except fo the mddle tem, we get 1 ( ( A U(, x ( 1 B 1 + ( 1 α β x + A B 1 α β x ( A ( B + 1 ( b x ( A ( 1 B + A B (A B α β + A B α β x 1 (α β + α β x + α β x ( AB 1 ( b x 1 ( A ( 1 B + A B ( b (A B α + A B β x 1 ( b V x + b x ( ( AB + 1 ( b x. If U 0 0, then A B U 1 α β, and n ths case we can deve the followng beautful denttes 3

4 Theoem. We have U(, x A ( U(, x A ( 1 ( b U x 1 ( b V x b, f odd x ( b V x 1 ( b V x + b x + ( ( 1 A, f even. 1 ( b x Coollay 3. If {U n } n s a non-degeneate second-ode ecuence sequence and U 0 0, then U(1, x U(, x U(3, x U 1 x 1 ax bx (1 U1 x(1 bx (bx + 1(b x ( V x + 1 δa U 1 x ( 1 abx b 3 x (1 V 3 x b 3 x (1 + bv 1 x b 3 x. (3 Poof. We use Theoem. The fst two denttes ae staghtfowad. Now, ( U(3, x A U 3 x 1 V 3 x b 3 x + ( 3 1 bu 1 x 1 + bv 1 x b 3 x A x U 3 + 3bU 1 + b(u 3 V 1 3U 1 V 3 x b 3 (U 3 + 3bU 1 x (1 V 3 x b 3 x (1 + bv 1 x b 3 x δa U 1 x ( 1 abx b 3 x (1 V 3 x b 3 x (1 + bv 1 x b 3 x, snce U 3 + 3bU 1 (a + 4bU 1 δu 1 and U 3 V 1 3U 1 V 3 aδu 1. Rema 4. If U n F n, the Fbonacc sequence, then a b 1, and f U n P n, the Pell sequence, then a, b 1. 3 Hoadam s Theoem Hoadam [3] found some closed foms fo patal sums S n P, S n P, whee P n s the genealzed Pell sequence, P n+1 P n + P n 1, P 1 p, P q. Let p n be the odnay Pell sequence, wth p 1, q, and q n be the sequence satsfyng the same ecuence, wth p 1, q 3. He poved 1 1 4

5 Theoem (Hoadam. Fo any n, S 4n q n (pq n 1 + qq n + p q; S 4n q n 1 (pq n + qq n 1 S 4n+1 q n (pq n + qq n+1 q; S 4n q n ( pq n+ + qq n+1 + 3p q; S 4n 1 q n (pq n + qq n 1 p S 4n+ q n ( pq n + qq n 1 + p S 4n+1 q n (pq n+1 qq n + p; S 4n 1 q n+1 (pq n+ qq n+1 + p q. We obseve that Hoadam s theoem s a patcula case of the patal sum fo a nondegeneate second-ode ecuence sequence U n. In fact, we genealze t even moe by fndng Sn,(x U U x. Fo smplcty, we let U 0 0. Thus, U n A(α n β n and V n α n + β n. We pove Theoem 6. We have S U n,(x A 1 x f s odd, and ( b U ( b n U ( (n+1 x n + ( b +(n 1 U ( n x n+1 1 ( b V x b x, (4 1 S U n,(x A ( 1 ( ( b (n+1 x n+1 1 ( b x 1 1 ( + A ( 1 ( b V x ( b (n+1 V ( (n+1 x n+1 + ( b +n V ( n x n+ 1 ( b V x + b x, ( f s even. Poof. We evaluate S U n,(x ( (Aα ( Aβ x ( A ( 1 (α β x ( (α A ( 1 β x n+1 1 α β. x 1

6 Assume odd. Then, assocatng, we get S U n,(x A A A ( ( (α ( 1 β x n+1 1 α β x (α β x n+1 1 α β x 1 ( (α ( 1 β x 1(α ( (n+1 β (n+1 x n+1 1 (α β x 1(α (n+1 β ( (n+1 x n+1 1 (α β x 1(α β x 1 ( α ( 1 (n+1 n β +n x n+ α ( (n+1 β (n+1 x n+1 1 α β x α +n β (n+1 n x n+ + α β x +α (n+1 β ( (n+1 x n+1 1 ( b (α + β x + α β x get A ( ( b ( 1 (α β x ( b (n+1 (α ( (n+1 1 A 1 x 1 β ( (n+1 x n+1 + ( b +n (α ( n β ( n x n+ 1 ( b V x b x ( b U ( b n U ( (n+1 x n + ( b +(n 1 U ( n x n+1 1 ( b V x b x. Assume even. Then, as befoe, assocatng, except fo the mddle tem, we 1 ( ( b Sn,(x U A ( 1 (α + β x ( b (n+1 (α ( (n+1 +β ( (n+1 x n+1 + ( b +n (α ( n + β ( n x n+ ( +A ( 1 ( A ( 1 ( b (n+1 x n+1 1 ( b x 1 1 ( b V x + b x ( b (n+1 x n+1 1 ( b x 1 1 ( + A ( 1 ( b V x ( b (n+1 V ( (n+1 x n+1 + ( b +n V ( n x n+ 1 ( b V x + b x. 6

7 Tang 1, we get the patal sum fo any non-degeneate second-ode ecuence sequence, wth U 0 0, Coollay 7. S U n,1(x x ( U 1 U n+1 x n bu n x n+1 1 V 1 x bx Rema 8. Hoadam s theoem follows easly, snce S n S P n,1 (1. Also S n can be found wthout dffculty, by obsevng that P n pp n + qp n 1 p( 1 n+ p n+ q( 1 n+1 p n+1, and usng S p n,1 ( 1. 4 Weghted Combnatoal Sums In [6] thee ae qute a few denttes le ( n F F n, o ( n F, whch s [ n 1 ] L n f n even, and [ n 1 ] F n, f n odd. A natual queston s: fo fxed, what s the closed fom fo the weghted sum n F (f t exsts? We ae able to answe the pevous queston, not only fo the Fbonacc sequence, but also fo any second-ode ecuence sequence U n, ( n n a moe geneal settng. Let S,n (x U x. Theoem 9. We have S,n (x Moeove, f U 0 0, then S,n (x A Poof. Let S,n (x ( A ( B (1 + α β x n. ( ( 1 (1 + α β x n. ( ( n (Aα ( Bβ x ( ( n A ( B (α β x ( A ( B (1 + α β x n 7

8 If U 0 0, then A B, and S,n (x A ( 1 ( (1 + α β x n Although we found an answe, t s not vey exctng. Howeve, by studyng Theoem 9, we obseve that we mght be able to get nce sums nvolvng the Fbonacc and Lucas sequences (o any such sequence, fo that matte, f we ae able to expess 1 plus/mnus a powe of α, β as the same multple of a powe of α, espectvely β. When U n F n, the Fbonacc sequence, the followng lemma does exactly what we need. Lemma 10. The followng denttes ae tue α s ( 1 s α s F s β s ( 1 s β s F s α s + ( 1 s L s α s (6 β s + ( 1 s L s β s. Poof. Staghtfowad usng the Bnet fomula fo F s and L s. Theoem 11. We have ( S 4+,n (1 n (+1 F+1 n F n(+1, f n odd (7 ( S 4+,n (1 n 4 + (+1 ( 1 F+1 n L n(+1 f n even (8 [ 1 ( ( ] 4 4 S 4,n (1 ( 1 (n+1 L n L ( n + n. (9 Poof. We use Theoem 9. Assocatng 4 +, except fo the mddle tem n S 4+,n (1, we obtan ( 4 + [( S 4+,n (1 (+1 ( α β 4+ n + (1 + α 4+ β n] ( 4 + [( (+1 ( ( 1 β 4+ n + (1 + ( 1 α 4+ n] 8

9 ( 4 + [( (+1 ( 1 (n+1 ( 1 + β (+1 n + (( 1 + α (+1 n]. We dd not nset the mddle tem, snce t s equal to ( 4 + (+1 ( 1 +1 (1 + α +1 β +1 n + 1 ( 4 + (+1 ( 1 +1 (1 + ( 1 +1 n In (10, usng (6, and obsevng that α (+1 + ( 1 α (+1 ( 1 +1, we get ( 4 + ( S 4+,n (1 (+1 ( 1 (n+1 n F n+1 ( 1 n β n(+1 + α n(+1. Theefoe, f n s odd, then and, f n s even, then ( 4 + S 4+,n (1 (+1 n+1 F n +1 F n(+1 ( 4 + S 4+,n (1 (+1 ( 1 n F n +1 L n(+1. In the same way, assocatng 4, except fo the mddle tem, and usng Lemma 10, (10 we get 1 ( 4 [( S 4,n (1 ( α β 4 n + (1 + α 4 β ( n] 4 + n 1 ( 4 [( ( 1 (n+1 ( 1 + β ( n + (( 1 + α ( n] ( 4 + n [ 1 ( 4 ( ( ] 4 ( 1 (n+1 L n β( n + L n α( n + n [ 1 ( ( ] 4 4 ( 1 (n+1 L n L ( n + n. (11 9

10 Rema 1. In the same manne we can fnd ( n U px. We now lst some nteestng specal cases of Theoems 9 and 11. Coollay 13. We have n F F n F n 1 L n F n F n+1 F 3 1 (n F n + 3F n ( n F 4 1 (3n L n 4( 1 n L n + 6 n. Poof. The second, thd and ffth denttes follow fom Theoem 11. Now, usng Theoem 9, wth A 1, we get S 1,n (1 1 1 ( 1 ( 1 1 (1 + α β 1 n 1 ( (1 + β n + (1 + α n 1 (α n β n F n. Next, the fouth dentty follows fom S 3,n (1 1 3 ( 3 ( 1 3 (1 + α β 3 n 1 [ (1 + β 3 n + 3(1 + αβ n 3(1 + α β n + (1 + α 3 n] 1 [ (β n + 3α n 3β n + (α n] 1 (n F n + 3F n, snce 1 + β 3 β, 1 + α 3 α. The esults n ou next theoem ae obtaned by puttng x 1 n Theoem 9, and snce the poofs ae smla to the poofs n Theoem 11, we omt them. 10

11 Theoem 14. We have 1 ( S 4,n ( 1 n 4 ( 1 F n L ( n, f n even, 1 ( S 4,n ( 1 n+1 4 F n F ( n, f n odd, [ ( ( ] S 4+,n ( 1 (+1 ( 1 (n+1+n L n +1 L (+1 n n. + 1 Next we ecod some nteestng specal cases of Theoem 9 and 14. Coollay 1. We have ( 1 ( 1 ( 1 ( 1 ( 1 F F n F 1 ( ( 1 n L n n+1 F 3 1 (( n F n 3F n F 4 n 4 (L n 4L n, f n even F 4 n 3 (Fn + 4F n, f n odd. Poof. The fst dentty s a smple applcaton of Theoem 9. The denttes fo even powes ae mmedate consequences of Theoem 14. Now, usng Theoem 9, we get S 3,n ( 1 1 ( (1 β 3 n + 3(1 αβ n 3(1 α β n + (1 α 3 n 1 ( ( n β n + 3β n 3α n + ( n α n 1 (( n F n 3F n, snce 1 β 3 β, 1 α 3 α. Fom (9 we obtan, fo 1, 1 ( 4 ( 1 (n+1 ( 4 L n L ( n + n 11 0 (mod.

12 Smla conguences esults fom othe sums n Secton 4, and we leave these fo the eade to fomulate. Acnowledgements. The autho would le to than the anonymous efeee fo he/hs helpful and vey detaled comments, whch mpoved sgnfcantly the pesentaton of the pape. Refeences [1] L. Caltz, Geneatng Functons fo Powes of Cetan Sequences of Numbes, Due Math. J. 9 (196, pp [] A.F. Hoadam, Geneatng functons fo powes of a cetan genealzed sequence of numbes, Due Math. J. 3 (196, pp [3] A.F. Hoadam, Patal Sums fo Second-Ode Recuence Sequences, Fbonacc Quately, Nov. 1994, pp [4] J. Rodan, Geneatng functons fo powes of Fbonacc numbes, Due Math. J. 9 (196, pp. -1. [] M. Rumney, E.J.F. Pmose, Relatons between a Sequence of Fbonacc Type and a Sequence of ts Patal Sums, The Fbonacc Quately, 9.3 (1971, pp [6] S. Vajda, Fbonacc & Lucas Numbe and the Golden Secton - Theoy and Applcatons, John Wley & Sons, AMS Classfcaton Numbes: 11B37,11B39, 0A10, 0A19 1