Fibonacci and Some of His Relations Anthony Sofo School of Computer Science and Mathematics, Victoria University of Technology,Victoria, Australia.

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1 The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe Fiboacci ad Some of His Relatios Athoy Sofo School of Compute Sciece ad Mathematics, Victoia Uivesity of Techology,Victoia, Austalia Abstact: I this aticle we evisit the Fiboacci sequece ad exted it i vaious diectios We coted this is a good topic to get people iteested i ecueces ad closed fom epesetatio We discuss vaious Fiboacci elated sequeces ad fially epeset two thid ode ecuece elatios i tems of biomial sums ad hypegeometic fuctios Itoductio Leoado Fiboacci was a mathematical iovato of the thiteeth cetuy He was bo i Pisa, Italy ad was also kow as Leoado Pisao, o Leoado of Pisa Fiboacci had a Mooish schoolmaste, who impotatly itoduced him to the Hidu-Aabic umeatio system ad computatioal methods Abacus is a wod that has had seveal meaigs i the past It is still used today to deote the Japaese ad Chiese computig devices sooba ad sua pa Fiboacci uses it i a way peculia to him, he gives it the sese of mathematics with the exclusio of Geomety I cetai cotexts, he uses the wod NUMERUS, that is, umbe, also i this sese Thus libe abbaci of libe de umeo simply meas book of mathematics Libe abbaci, witte i, was a book vey ahead of its time, its level was ot eached agai util thee huded yeas late by Luca Pacioli s Summa de Aithmetica, Geometia, Popotioi & Popotioalita (Veice, 59), i which Pacioli wites explicitly that he esumes the wok of Fiboacci Libe abbaci, was witte by Fiboacci afte widespead tavel ad extesive study of computatioal systems I it, he explais the Hidu-Aabic umeals ad how they ae used i computatio This extemely famous book was istumetal i displacig the clumsy Roma umeatio system ad itoduced methods of computatio simila to those used today Fiboacci is today, paticulaly emembeed fo the sequece of umbes {,,,,, 5, 8,,, 4, 55, 89, 44, } that is associated with the beedig of abbits Rabbits I the libe abbaci [], Fiboacci itoduced the followig abbit stoy which geeated his famous sequece Suppose that thee is oe pai of abbits i a eclosue o the fist day of Jauay, this pai will poduce aothe pai of abbits o Febuay fist ad o the fist day of evey moth theeafte; ad each ew pai will matue fo oe moth ad the poduce a ew pai o the fist day of the thid moth of its life ad o the fist day of evey moth theeafte The poblem is to fid the umbe of abbits i the eclosue o the fist day of the followig Jauay afte the biths have take place o that day Let A deote a adult pai of abbits ad let B deote a baby pai of abbits, the followig descibes the umbes afte oe yea Total umbe Moth Numbe of A s Numbe of B s of pais Jauay Afte biths o fist of Febuay Mach Apil 5 May 5 8 Jue 8 5 July 8 August 4 57

2 The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe Septembe 4 55 Octobe Novembe Decembe Jauay Theefoe the umbe of pais of abbits i the eclosue oe yea late would be 77 I geeal, fom the last colum, we may see that the umbe of pais at ay moth is the sum of the two peceedig eties which leads us to the idea of a ecusio fomula Recueces ad Sums A sequece may be fiite o ifiite ad may be desigated by symbols such as {a, u, u,, u, } A ecuece (o diffeece) equatio is the discete aalog of a diffeetial equatio ad may be epeseted by f ( ) f ( ) y( ), whee is a atual umbe The Fiboacci ecuece fomula may be witte as F F + F, > ( R) with the iitial values F, F We ow biefly descibe the followig sequeces ad seies, some of which will be useful late A aithmetic sequece is A : { a, a + d, a + d +L} hee a is the fist tem ad d is the diffeece betwee + tems If SA : ( a + id ), the SA : ( a + d ) i A geometic sequece is G : { a, a, a, L, a } If i G a, whee is the costat atio of two i subsequet tems, the a( ) G Fo a ifiite umbe of tems a lim G fo < Biomial sums ae impotat ad ae itisically coected to hypegeometic fuctios ( x y) x y + is a Biomial sum, whee! ad! ( ), K,,,! ( )! T + If the atio of two cosecutive tems, i a seies T, is a atioal fuctio of a positive itege the T we have a hypegeometic seies a ( ) ( ), a, K, a a a p p z p Fq z, b, b,, b L K p ( b ) L( bq )! a is Pochhamme s fuctio defied by whee T ad ( ) 58

3 The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe ( a) ( a) a( a + ) L( a + ) Γ( a + ) Γ( a) ad Γ ( x) is the classical Gamma fuctio Now we etu to the Fiboacci ecuece ad ivestigate some of its popeties Popeties The Fiboacci sequece has may emakable popeties, moeove, it cotiues to fid applicatios i may aeas of sciece ad mathematics Hagspihl [] oted fo the Fiboacci umbes, 5, 8, that ( ) + ( 5 8) 89 ; ( F F ) + ( F F ) ( F ), that is, the Fiboacci tiagle equality I geeal, this popety may be poved fo ay fou subsequet Fiboacci umbes Coside F, F +, F +, F+ the ( F F+ ) + ( F+ F+ ) F ( + ) ( P) To pove (P), we may wite fom (R), ( )( ) + F + F + F + + F + F + F F F Fom (P), ( ) F F + F F F + F F + F ( F + F ) F ( ) As oted by Hagspihl, fo a ight agled tiagle with the two shote sides as ( F F + ) ad ( F + F ) the the aea is equal to F F+ F+ F+ The Fiboacci ecuece elatio (R) has the chaacteistic equatio x x ( x α )( x β ), + which poduces two oots, whee α the golde atio ad β ( P) α β Usig these two oots we obtai the classical Biét fom F α β The Fiboacci sequece {F } may also be witte i tems of a sum such that [ / ] / F + + L +, ( P) / whee [x] is defied as the geatest itege less tha o equal to x, (the floo fuctio) The Fiboacci umbes may also be witte as the poduct of tigoometic fuctios, F i cos ( P4) + Oe of the most impotat popeties of the Fiboacci umbes, kow as Zeckedof s theoem [8] is the special way i which they ca be used to epeset iteges Evey positive itege has a uique epesetatio of the fom F + F + L + F, whee >> >> L >> >>,, F + F6 + F4 + F + F 8,4 +,9 + 46, Relatio A moe geeal elatio is the geealized Fiboacci polyomials g + bg cg, g, ( P5) whee b ad c ae eal umbes, fom which we obtai the polyomials (see diagam)

4 The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe The ecuece elatio (P5) is moe geeal tha (R) because fo vaious values of b ad c we may obtai the Jacobsthal, Pell, Femat, Čebyšev ad Fiboacci sequeces The chaacteistic equatio of (P5) is x bx c fom which we obtai the two oots b + b + 4c b b + 4c ξ ad ξ g b b + c b + bc The sequece { g } may the be expessed as [ / ] ξ ξ b + b + c b b + c g c b + ξ ξ b c May othe idetities of this fom may be obtaied, fo example utilizig (P4) we may coclude that π ( + ) si 4 + Padoa ad Pei The Padoa itege sequece, see Sloae [5] ad Stewat [7], is defied by the ecuece elatio P P P ; ( P6 with iitial coditio P ( ) P( ) P( ) The chaacteistic equatio of (P6) is with oots ( ) ( ) ( ) ) x x ρ , ρ i ρ ρ b + b b + 4b c + bc 5 5 b + 5b c + 6b b + 6b c + b c + 4bc b + 7b c + 5b c + b b + 8b c + b c + b c + 5bc b + 9b c + 8b c + 5b c + 5b whee ρ is the complex cougate of ρ The closed fom epesetatio of (P6) ca be witte as ad ( P7) 6

5 The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe + p P ( ) ( ) + p + p The Pei itege sequece, see Sloae [5] ad Pei [4], is defied by (P6) but with the iitial P, P, P It has the same chaacteistic equatio (P7) ad i this case the coditios ( ) ( ) ( ) p closed fom epesetatio of the Pei sequece is ( ) limitig atio of the successive tems, ad theefoe P The plastic costat is defied as the ( + ) P( ) P lim ρ Sofo [6] has exteded the Padova ad Pei itege sequeces by cosideig the ecuece elatio S ( + ) + S( ) S( ) ( P8) with iitial coditios S ( ), S( ) ad S( ) 4 Hece we obtai the alteatig sig tems { S ( ) } {,, 4, 7,,,, 54, 88, 4,, K} The chaacteistic equatio of (P8) is x + x with oots x, x α ad x β, whee α ad β ae the classical Fiboacci umbes (P) I tems of biomial sums ad hypegeometic fuctios we may wite + + [ / ],, 7 S( ) ( ) ( ) F ( P9) + The plastic costat + β 5 α ( ) + S lim ( + ) S( ) α, + α β the golde atio A iteestig expessio of the ifiite fom of (P9) ca be witte as, see Sofo [6] + α ( ) ( ) S α 4 Not all sequeces ae ecessaily easy to epeset as a ecuece elatio o i closed fom Coside the sequece { R( ) } {,,,,, 5,,,, 9,,, 5, 77, 5, 5, 5, 7, 5, 5, 89, 9, 89, K}, at fist sight it looks itactable, i fact, we ca epeset it by the ecuece elatio R( + ) R( ) R( ) ( P) with iitial coditios R ( ) R( ) R( ) A close look at (P) idicates that it is simila to (P8) with the coefficiets of S ( ) ad S ( ) chaged to ad espectively The ecuece (P) has the chaacteistic equatio x x + with oots {, + i, i}, ad ca be epeseted i biomial ad hypegeometic fom as [ / ],, 7 ( ): ( ) + π π R F ( ) + Cos + Si,

6 The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe Some othe sums ad poducts ca be foud i a aticle by Bowei ad Coless [] Refeeces [] Baldassae, B Scitti di Leoado Pisao, Vol I, Il libe abbaci, Roma, 857 [] Bowei, J & Coless, R Emegig poofs fo expeimetal mathematics, AmeMathMothly,6 (999), [] Haspihl, T It s ot supisig that Euclid got excited about geomety, p 58 6, Poceedigs of the Iteatioal Cofeece, The Humaistic Reaissace i Mathematics Educatio, Italy, [4] Pei, R Item 484 L Itemediae des Math, 6 (899), [5] Sloae, NJA Sequece A9 i The O-lie Ecyclopedia of Itege Sequeces [6] Sofo, A Computatioal Techiques fo the Summatio of Seies To be published by Kluwe Academic Publishes, New Yok, [7] Stewat, I Tales of a Neglected Numbe, Sci Ame, 74 (996), - [8] Zeckedof, E Repesetatio des ombes atuels pa ue somme de ombes de Fiboacci ou de ombes de Lucas, Bulleti de la Société Royale des Sciece de Liège, 4 (97),