LINEAR RECURRENT SEQUENCES AND POWERS OF A SQUARE MATRIX

Transcription

1 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 LINEAR RECURRENT SEQUENCES AND POWERS OF A SQUARE MATRIX Hacèe Belbachr 1 USTHB, Departmet of Mathematcs, POBox 32 El Ala, 16111, Bab Ezzouar, Algers, Algera hbelbachr@usthbdz ad haceebelbachr@gmalcom Fard Becherf 2 USTHB, Departmet of Mathematcs, POBox 32 El Ala, 16111, Bab Ezzouar, Algers, Algera fbecherf@usthbdz ad fbecherf@yahoofr Receved: 8/10/05, Revsed: 3/20/06, Accepted: 4/4/06, Publshed: 4/24/06 Abstract I ths paper, we establsh a formula expressg explctly the geeral term of a lear recurret sequece, allowg us to geeralze the orgal result of J McLaughl [7] cocerg powers of a matrx of sze 2, to the case of a square matrx of sze m 2 Idettes cocerg Fboacc ad Strlg umbers ad varous combatoral relatos are derved 1 Itroducto The ma theorem of J McLaughl [7] states the followg: a b Theorem 1 Let A be a square matrx of order two, let T a + d be ts trace, c d ad let D ad bc be ts determat Let y /2 0 T 2 D 1 The, for 1, A y dy 1 by 1 cy 1 y ay Partally supported by the laboratory LAID3 2 Partally supported by the laboratory LATN

2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 2 We remar that ths theorem, y 1 s a lear recurret sequece that satsfes y 1 0, y 0 1, y T y 1 Dy 2 for all teger d b By settg A 0 ad A c a wrtte as follows: 3, relato 2 of Theorem 1 may be A y A 0 + y 1 A 1 for all tegers 0, 4 wth A 0 I 2 ad A 1 A T I 2 where I 2 s the ut matrx I Secto 3, we exted ths result relato 4 to ay matrx A M m A of order m 2, A beg a utary commutatve rg We prove the followg result: Let A M m A ad let P X X m a 1 X a X a m be the characterstc polyomal of A Let A 0, A 1,, A be matrces of M m A defed by A a A, for 0 m 1, wth a 0 1, 0 ad let y > m be the sequece of elemets of A satsfyg 1 + y m a 1 1 a 2 2 a m m, 1, 2,, m m m The, for all tegers 0, A y A 0 + y 1 A y m+1 A The proof of ths result s based o Theorem 2 gve Secto 2 for > m I ths secto, we geeralze Theorem 2, whch permts us to express the geeral term u of a recurret lear sequece satsfyg the relato u a 1 u 1 + a 2 u a m u m for all 1, terms of the coeffcets a 1, a 2,, a m ad u 0, u 1, u 2,, u Applcatos to Fboacc, geeralzed Fboacc ad multboacc sequeces are also gve Fally, Secto 4, further combatoral dettes are derved, cludg dettes cocerg the Strlg umbers of the frst ad secod d As a llustrato, we gve a ce dualty betwee the two followg relatos Corollares 5 ad 7: 1 m m 1 m 1 m m + m 1, 1,, m m 1 0 m m m

3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 3 [ ] 1 [ ] m { m 1 m 1 m m + m 1 1,, m m 1 0 m m m [ ] where m s the multomal coeffcet Secto 2, ad ad 1,, m are, respectvely, the Strlg umbers of the frst ad secod d as defed [5] { }, } 2 Explct Expresso of the Geeral Term of a Recurret Lear Sequece I ths secto, we let m 2 be a teger, A a utary commutatve rg, a 1, a 2,, a m, α 0, α 1,, α elemets of A, ad u > m a sequece of elemets of A defed by { u j α j for 0 j m 1, 5 u a 1 u 1 + a 2 u a m u m for 1 The am of ths secto s to gve a explct expresso of u terms of, a 1, a 2,, a m, α 0, α 1,, α Theorem 3 Let us defe the sequece y Z of elemets of A, wth the coveto that a empty sum s zero, by 1 + y m a 1 1 a 2 2 a m m, for Z, 6 1, 2,, m m m the summato beg tae over all m-tuples 1, 2,, m of tegers j 0 satsfyg m m Wth the prevous coveto, we have y 0 for < 0 The multomal coeffcet that appears the summato s defed for tegers 1, 2,, m 0, by m m!, 1, 2,, m 1! 2! m! ad ca always be wrtte as a product of bomal coeffcets m m , 2,, m Let us adopt the followg coveto For m 1, we put j m 0 whe 1, 2,, j 1,, j 0, m for ay j {1, 2,, m} We ca ow state the followg lemma [p 80 Vol 1, 4] Lemma 1 Let j 0 be tegers for j {1, 2,, m}, such that m 1 The m m j m 1, 2,, m 1,, j 1,, m j1

4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 4 Ths lemma permts us to easly prove the followg result Lemma 2 The sequece y Z, defed by relato 6, satsfes the recurrece relato wth y 0 1 ad y 0 for < 0 y a 1 y 1 + a 2 y a m y m for 1 Proof Frst otce that, for 1, for all j {1, 2,, m} we have a j y j j m a 1 1 a j j a m m 1,, j 1,, m m m Applyg Lemma 1, we obta m a j y j y The relatos y 0 1 ad y 0 for < 0 follow mmedately j1 We ca ow state the followg result Theorem 2 Let u > m the sequece of elemets of A defed by u j 0 for 1 j m 1, u 0 1, u a 1 u 1 + a 2 u a m u m for 1 The for all tegers > m, u m m m 1, 2,, m a 1 1 a 2 2 a m m Corollary 1 Let q 1 be a teger, a, b A, ad let v q be the sequece of elemets of A defed by The, for all q, v j 0 for 1 j q, v 0 1, v +1 av + bv q for 0 7 ad, for all 0, v q+1 q a q+1 b, 8 v +1 + bv q 2v +1 av +1 q q q + 1 q a +1 q+1 b 9

5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 5 Proof We deduce relato 8 drectly from Theorem 2, wth m q + 1, a 1 a, a m b ad a 0 for 1 < < m From 8, we deduce 9 as follows: v +1 + bv q +1 q q a +1 q+1 b + +1 q+1 q q+1 +1 q q a +1 q+1 b + +1 q q q a +1 q+1 b + +1 q q q + 1 q We ow gve some applcatos of the above corollary + 1 q q 1 +1 q a +1 q+1 b Applcato 1 Let F 0 be the Fboacc sequece F 0 0, F 1 1, F +1 F + F 1 for 1 a q q+1 b +1 a +1 q+1 b + 1 q The, by settg ϕ F +1 for 1, we see that ϕ 1 s also defed by ϕ 1 0, ϕ 0 1, ϕ ϕ 1 + ϕ 2 for 1 The applcato of Corollary 1 gves us that ϕ F +1 /2, for 1, the relato gve [pp 18-20, 12], ad aouced [9] Also F + F , for 0, ad we fd the relatos gve Problem 698 of [10], whch state that 2 2 F F a +1 q+1 b

6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 6 ad F 2 + F Applcato 2 For q 1, let G q be the geeralzed Fboacc sequece as cted 0 [8], ad let be a sequece of umbers defed as follows: H q 0 { G q 0 G q 1 G q q 1, G q +1 G q + G q q for q, ad { H q 0 H q 1 H q q 1, H q +1 H q H q q for q We ca exted easly the above sequeces to G q ad H q by G q j 0 for 1 j q, G q 0 1, G q +1 G q + G q q for 0, ad H q j H q 0 1, The applcato of Corollary 1 gves us, for q, the relatos 0 for 1 j q, H q +1 H q H q q for 0 G q q+1 q, ad H q q+1 1 q Notce that G 1 F +1 ϕ, ad H 2 s the teger fucto studed by L Berste [1], who showed that the oly zeros of H 2 are at 3 ad 12 Ths result was treated also by L Carltz [2, 3] ad recetly by J McLaughl ad B Sury [8] The followg theorem gves us a explct formulato for u terms of, a 1, a 2,, a m, α 0, α 1,, α, ad thus geeralzes Theorem 2 Theorem 3 Let u > m be a sequece of elemets of A defed by { u j α j for 0 j m 1, u a 1 u 1 + a 2 u a m u m for 1 10 Let λ j 0 j ad y > m be the sequeces of elemets of A defed by λ j a j α, for 0 j m 1, wth a 0 1, 11 j ad y m m m 1, 2,, m a 1 1 a 2 2 a m m, for > m

7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 7 The for all teger > m, we have u λ 0 y + λ 1 y λ y + Remar Note that 11 s equvalet to λ 0 λ 1 λ 2 or λ m 2 λ 1 a 1 a 2 a m 2 a 0 1 a 1 a 2 a m a 1 a m a α 0 α 1 α 2 α m 2 α [λ 0, λ 1,, λ ] t C [α 0, α 1,, α ] t, 12 where 0 f > j, C c j 1,j m, wth c j 1 f j, a j f < j We deduce also from relatos 10 ad 12 the matrx equalty u y y +1 y +m 2 y + u +1 y +1 y +2 y + y +m u +m 2 y +m 2 y + y +2m 4 y +2m 3 u + y + y +m y +2m 3 y +2m 2 1 a 1 a 2 a 0 1 a 1 a a α 0 α 1 α m 2 α Proof Let S be the A-module of sequeces v > m satsfyg the recurrece relato v a 1 v 1 a 2 v 2 a m v m 0 for all 1 Let us cosder the famly of sequeces, for 0 m 1, defed by v 0 y, v 1 v 2 y +1 a 1 y, v > m y +2 a 1 y +1 a 2 y, v y + a 1 y +m 2 a 2 y +m 3 a y,

8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 8 e, v y + a 1 y a 2 y a y 13 a y +, wth a By Theorem 2, we have y > m S Cosequetly, y +q > m S for q 0, ad fally we deduce that v S, for 0 m 1 14 > m It s easy to observe that we also have, for j, {0, 1, 2,, m 1}, v j δ j { 1 f j 0 f j 15 I fact, wth a 0 1, we ca wrte, for j, {0, 1,, m 1},v j 0 a y j+ If < j, the j + < 0 for 1 ad v j 0 because y q 0 for q < 0 If j, the v j vj j j a y a 0 y 0 1 If > j, the for r j < 0, we have 0 r 1 ad v j y r a 1 y r 1 + a 2 y r a y r, ad by usg r j 0, y r a 1 y r 1 + a 2 y r a y r + + a m y r m 0 because y > m S ad r 1 Relatos 14 ad 15 gve easly, that for all > m,u u a y + α 0 a j α y +j j0 j0 j0 j λ j y +j, a j α y +j α v ad, wth 13, where λ j s as defed 11

9 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 9 Theorem 3 allows us to fd varous formulae for the Fboacc umbers Corollary 2 For all tegers 1, F Remar I ths summato, we may, fact, restrct the sum to those tegers ragg betwee +1, the bomal coeffcet of the formula beg zero for the other tegers ad Proof Note that F 2F 2 +F 3 for 3 Deotg by y 2 the sequece defed by 2+3l y 2 y 1 0, y 0 1, y 2y 2 + y 3 for 1, we see that, for 1,F y 1 + y 2 Theorem 3 allows us to state that, for 2, y + l 2 t 2 3t, l 2t 0 t ad Corollary 2 follows by smple calculatos The followg result ca also be easly deduced from Theorem 3 Corollary 3 For all tegers m 1, F 2 m/3 m + m m 2 Now, let us cosder for q > 1, the multboacc sequece φ q q 1 φq 2 φ q 1 0, φ q 0 1, φ q φ q φ q 1 + φ q φ q q for 1, 2 3 > q defed by 16 where φ 2 ϕ F +1 G 1 Theorem 3 also mples that, for all 0, φ q q q q 1, 2,, q Thus, for q 3, we obta φ 3 +2j+3 + j +, j, 2+3j 2j + j + j

10 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 10 We deduce from 16 that φ q +1 2φ q φ q, q for 1 Let us cosder ψ q, the sequece defed by ψ : φ q, for > q ad ψ q 1, whch satsfes the recurrece relatoψ +1 2ψ ψ q, for 0 After applyg Theorem 3, we fd that, for 0, wth z q+1 q ψ φ q λ0 z + λ 1 z λ q z +q z 2z +q 1 + z +q, 2 q+1 1, for q We ow, va Theorem 3 ad Corollary 1, that the sequece z q satsfes the recurrece relato z +1 2z + z q 0, for 0 Ths gves, for 0, φ q z z 1 + z +q 2z +q 1 + z 1 z z 1 Applyg relato 9 Corollary 1, we ca wrte, for 1, Thus, q q φ q q q 1,, q q 1 q q+1 q q 1 q 2 1 q+1 1 q 2 1 q For q 3, we obta + j +, j, +2j+3 2+3j 2j + j + j Powers of a Square Matrx of Order m We start ths secto wth the ma result of ths paper Theorem 4 Let A M m A ad let P X X m a 1 X a X a m be the characterstc polyomal of A Let A 0, A 1,, A be matrces of M m A defed by A a A, for 0 m 1, wth a 0 1, 18 0

11 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 11 ad let y > m y be the sequece of elemets of A satsfyg m m m 1, 2,, m a 1 1 a 2 2 a m m, for > m The, for all tegers 0, A y A 0 + y 1 A y m+1 A 19 Proof Defe, for 0 m, For 0 m 1, we have P X a X, wth a XP X P +1 X + a Relato 20 shows that the degree of P s for 0 m It mples that P 0, P 1,, P s a bass of the free A-module A [X] cosstg of polyomals of A [X] of degree For all 0, the remader R of the eucldea dvso of X by P m s wrtte as a lear combato of polyomals P 0, P 1,, P The, for all 0, there exsts a uque famly α, 0 such that R α, P 22 For 0, we have R X X, where X s a lear combato of P 0,, P, ad { α0, α, 0 for < m 1 Relatos 21 ad 22 mply that XR X α, P +1 X + a +1 a +1 α, + 1 α, 1 P X + α, P m X As a cosequece, the polyomal R +1 X XR X α, P m X, of degree m 1, s dvsble by P m X, whch s of degree m Ths polyomal s thus the zero polyomal as

12 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 12 well as ts compoets the bass P 0, P 1,, P Ths provdes us wth the followg relatos: α +1,0 a +1 α,, for 0, 24 α +1, α, 1 for 1 m 1 25 Let us set, for all tegers Z, z { α,0 for 0, 0 for < 0 26 Oe checs easly that for all tegers 0 ad for 0 m 1, α, z 27 Ideed, f ths relato follows from 25 ad 26, ad f 0 < m 1 t follows from 23 ad 26 From 25, 26 ad 27, we fd that z 0 for < 0, z 0 1, z a 1 z 1 + a 2 z a z + a m z m for 1 Theorem 2 mples that z y, for all Z, where 1 + y m 1, 2,, m m m a 1 1 a 2 2 a m m Ths last fact, together wth the Cayley-Hamlto Theorem, 22, 20, 18 ad 27 ow gve that A R A whch completes the proof of 19 0 α, P A 0 z A 0 y A 4 Further Combatoral Idettes Some ce combatoral dettes ca be derved from Theorem 4 by cosderg varous partcular matrces wth smple forms Corollary 4 Let be a postve teger The m m m 1,, m m m m 1 m 0 m + m 1 m 1,

13 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A m 1 where s the umber of m- combatos wth repetto of fte set {1,, }, m 1 ad the summato s tae over all m-tuples 1, 2,, m of tegers j 0 satsfyg the relato m m Proof Let J m be the m m Jorda matrx, J m The characterstc polyomal of J m s X 1 m We also have J m j 1,j m Applyg Theorem 4 wth A J m, ad cosderg the 1, m etres of both sdes of 19, we obta the relato y m 1, whch leads to the result From Corollary 4, we obta the followg combatoral dettes, for m 2, 3, , 2+3j+4 2+3j 2 1 2j + j 1 + 2j 3 + j + + j + j + + j 3 3j + j j Le J Mc Laughl ad B Sury [8, Corollary 6], we ca also derve Corollary 4 from the followg result Corollary 5 [8, Theorem 1] Let x 1, x 2,, x m be elemets of the utary commutatve rg A wth s x 1 x 2 x, for 1 m The, for each postve teger, m 1 1 < 2 < < m x 1 1 x m m m m m 1,, m 1 1 m s 1 1 s m m, wth the summatos beg tae over all m-tuples 1, 2,, m of tegers j 0 satsfyg the relatos m for the left-had sde ad m m for the rght-had sde

14 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 14 Proof Let us gve aproof of ths result by usg Theorem 2 For > m ad 1 l m, cosder q l : x 1 1 x 2 2 x l l, ad q : q m Let E : β β +1 be the shft l operator whch acts o ay sequece β, ad, for 1 l m, let Q l be the operator gve by Q l : E x 1 E x 2 E x l Notce that Q m E m s 1 E m s m The, for 1 ad 2 l m, we have E x l q l l ql 1 l 1 Therefore, Thus, Q m q m m Q q Q 1 q q s 1 q 1 s 2 q s m q m, for 1 28 By the defto of q, we also have q 0 for < 0 ad q Applyg Theorem 2 to the sequece q > m, we obta the result Corollary 4 follows mmedately by settg x 1 for all the prevous corollary The followg theorem s a exteso of a result of J Mc Laughl ad B Sury [8, Theorem 3] Theorem 5 Let K be a feld of characterstc zero, ad let x 1, x 2,, x m be depedet varables The, K x 1, x 2,, x m, m x 1 1 x 2 2 x m m m 1 x + j x x j Proof For γ 1, γ 2,, γ m K x 1, x 2,, x m, let V γ 1, γ 2,, γ m deote the Vadermode determat defed by V γ 1, γ 2,, γ m det γ j 1 It s well ow that 1,j m V γ 1, γ 2,, γ m γ j γ By 28, the sequece q > m defed by 1 <j m q m s a recurret sequece wth characterstc polyomal x 1 1 x 2 2 x m m X m s 1 X m s m X x 1 X x 2 X x m Ths polyomal has m dstcts roots x 1, x 2,, x m We deduce that there exst m elemets A A x 1, x 2,, x m K x 1, x 2,, x m, 1 m, such that q m A x 1 for m

15 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 15 The tal codtos gve by 28 lead to the Cramer system m 1 The resoluto of ths system gves V x 1,, x 1, 0, x +1,, x m A 1 1 V x 1,, x 1, x, x +1,, x m m 1 A x j δ j,0 for 0 j x j x, for 1 m x j Ths completes the proof Let us ow gve a applcato to Strlg [ ] umbers For 0, wth the otatos of [5], the Strlg umbers of the frst d, ad the Strlg umbers of the secod d { }, ca be defed by the equatos wth X : wth X : X [ { 1 f 0, X X + 1 X + 2 X + 1 f 1, X { { 1 f 0, X X 1 X 2 X + 1 f 1 It s well ow [p 38 Vol 2, 4] that ad Corollary 5, we deduce m m m 1,, m { ] X, 30 ad } X, 31 } 1! 1 From Theorem m s 1 1 s m m If we tae x 1 for 1 m, we have s 1 [ ad relato 32 gves the followg result m 1 m m x + j x x j 32 ], for 1 m, Corollary 6 For all postve tegers m ad, m m m 1,, m [ m m m 1 ] 1 [ m 0 ] m { + m 1 m 1 }

16 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 16 Note that Theorem 5 gves the followg relato, whch s stated [p 42, Vol 2, 4] { } + m 1 For all postve tegers m ad, m 1 m For m 3, 4 ad 5, we obta , 2+3j 1 2j + j + j 6 2 2j , 2 2+3j j 3 + j + + j + + j + j 2 2 2j 3 5 j Corollary 7 Let be a postve teger ad let x ad y be determates The 1 + 2j 3 + j + + j 2x + 2y 2 2j 4 + j + + j 2+3j+4 xy j+2 x 2 + 4xy + y x +3 y xy x +1 y +1 x y 3 Corollary 8 Let a postve teger ad x, y be determates The 1 + 2j 3 + j + + j + j + + j 2+3j+4 3x + y 2 3j 4 x +2j+3 y 3x + 3y x + 3y j x+3 y y [ x +2 y x +1 x y] 2 x y 3 Proof of Corollares 7 ad 8 It suffces to put, Corollary 4, x 1 x 2 x ad x 3 x 4 y to obta Corollary 7; x 1 x 2 x 3 x ad x 4 y to obta Corollary 8 Acowledgmets The authors are grateful to the referee ad would le to tha hm/her for commets whch mproved the qualty of ths paper

17 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY , #A12 17 Refereces [1] L Berste: Zeros of the fucto f J Number Theory, , [2] L Carltz: Some combatoral dettes of Berste SIAM J Math Aal, , o 1, [3] L Carltz: Recurreces of the thrd order ad related combatoral dettes Fboacc Quarterly, , o 1, [4] L Comtet: Aalyse combatore Puf, Coll Sup Pars, 1970, Vol 1 & Vol 2 [5] R L Graham, D E Kuth, O Patash: Cocrete Mathematcs Addso Wesley Publshg Compay, Ic, 1994 [6] R C Johso: Matrx methods for Fboacc ad related sequeces wwwduracu/ bobjohso/ fboacc/, 2004 [7] J Mc Laughl: Combatoral dettes dervg from the -th power of 2 2 matrx Electroc Joural of Combatoral Number Theory, A19, [8] J Mc Laughl, B Sury: Powers of matrx ad combatoral dettes Electroc Joural of Combatoral Number Theory, A13, [9] Rajesh Ram: Fboacc umber formulae fboacc/ 2003 [10] Z F Starc: Soluto of the Problem 698 Uv Beograd Publac Eletroteh Fa Ser Mat , [11] S Tay, M Zuer: Aalytc methods appled to a sequece of bomal coeffets Dscrete Math, , [12] N N Vorob ev: Fboacc umbers Traslated from the russa by Hala Moss, Traslato edtor Ia N Seddo Blasdell Publshg Co, New Yor-Lodo, 1961