ON SOME RELATIONSHIPS AMONG PELL, PELL-LUCAS AND MODIFIED PELL SEQUENCES

Transcription

1 SAÜ Fe Bilimleri Dergisi, Cilt, Sayı, s-5, 00 O Some Relatioships Amog ell, ell-lucas ad Modified ell Seueces ON SOME RELATIONSHIS AMONG ELL, ELL-LUCAS AND MODIFIED ELL SEQUENCES, Ahmet DAŞDEMİR Sakarya Uiversity, Scieces Ad Arts Faculty, Departmet of Mathematics, Esetepe Campus, Sakarya, Turkey shalici@sakaryaedutr ABSTRACT I this study, ell, ell-lucas ad Modified ell umbers are ivestigated Usig Biet formulas for these seueces, some relatioships amog these seueces are obtaied Also, some sum formulas are give by these properties Key words: ell Numbers, ell-lucas Numbers, Modified ell Numbers AMS Subject Classificatio: B37; B39 ELL, ELL-LUCAS VE MODIFIED ELL DİZİLERİ ARASINDA BAZI İLİŞKİLER ÖZET Bu çalışmada ell, ell-lucas ve Modified ell sayıları çalışıldı Bu sayı dizileri içi taımlaa Biet formülleri kullaılarak, bu dizileri birbirleriyle ola bazı ilişkileri ortaya kodu Bulua bu özellikler yardımıyla da bazı toplam formülleri verildi INTRODUCTION The Fiboacci ad Lucas umbers ad their geeralizatios very importat properties ad applicatios to almost every fields of sciece ad art The applicatios of these umbers ca be see i [7] Some seueces, such as ell seueces, a similar structure with the Fiboacci seuece [,, 3,, 6] ell seuece, { }, ca be defied as + ; + with iitial coditio 0 0, Moreover, the ell seueces ca be explaied by matrices I [], Ercolaa gave a matrix method for geeratig the ell seuece as follows; + 0 Usig this matrix, the followig euatio ca be writte: ( +

2 SAÜ Fe Bilimleri Dergisi, Cilt, Sayı, s-5, 00 O Some Relatioships Amog ell, ell-lucas ad Modified ell Seueces It is see that there is may relatioships betwee the matrices ad ell umbers These relatios ca be see i [, 3, 5] ( Q (, ell-lucas seuece ca be defied as Q Q + Q ; + where Q0 ; Q Also, Modified ell seuece { } ca be defied by the followig recursive relatio: +, + where 0 ad I [], Melham gave Biet formulas for the ell ad ell-lucas umbers ; α β, Q α β I [], Horadam gave Biet formula for Modified ell seuece, where α, x x 0 β are the roots of the euatio I this paper, we ivestigated ell, ell-lucas ad Modified ell umbers Also, we derive some miscalleous relatios by usig their Biet formulae MAIN RESULTS Now, we will give the followig lemma without proof However, the proof ca be easily obtaied usig the followig euatio + ( Q ( + Lemma If, Q, are th ell, ell-lucas ad Modified ell umbers, the for all positive itegers, we, ad + ( Q+ + ( + + ropositio If, Q, are th ell, ell-lucas ad Modified ell umbers, the for all positive itegers,m,k we + k+ ( ( + m+ k Q + m+ k + Qmk roof: Cosiderig the Biet formulas for ell, ell-lucas ad Modified ell umbers, we ca write + k+ ( ( Q + Q + m + m + k + k αβ αβ + m + k + m+ k mk k mk mk ( αβ ( αβ ( + m+ k + m+ k So, the proof is completed ropositio 3 If, are th ell ad Modified ell umbers, the for all positive itegers, we ( + + roof: Usig the Biet formulas of ell ad Modified ell, we get α β α β α + α β α β β By simple computatio, we obtai that ( α β(

3 SAÜ Fe Bilimleri Dergisi, Cilt, Sayı, s-5, 00 O Some Relatioships Amog ell, ell-lucas ad Modified ell Seueces αβ α α,,, α β, β β So, the proof is completed ropositio If, Q, are th ell, ell-lucas ad Modified ell umbers, the for all itegers, we ad i i+ +, i Q i i+ + i roof: Firstly, let us defie a ew seuece as follows; a + + ( From the defiitio of Biet formula for ell umbers, we ca write a + + ( ( α ( α + β ( β ( α β α α β β ( α β + + α β + + Now, usig the idea of creative telescopig [5], we coclude i i+ i i i a i i+ i i i+ ( i i i + + ( , which is desired ropositio 5 If is th ell umber, the we the followig euatio; 3

4 SAÜ Fe Bilimleri Dergisi, Cilt, Sayı, s-5, 00 O Some Relatioships Amog ell, ell-lucas ad Modified ell Seueces ( ( + + roof: Cosiderig the Biet formulas for ell umbers, we get + + ( + α ( α β ( β α β O the other had, we kow that α, β, we get α + β ( + Also, we obtai that α β + ( + ( α β ( αβ ( + ( ( + + ( α + β So, the proof is completed ropositio 6 If, Q, are ell, ell-lucas ad Modified ell umbers, the for all itegers, we Q Q Q + roof: If we use the Biet formlas of the ell ad Modified ell umbers ad αβ, the we α β α β α β α β + + α β α β By simple computatio, we get Ad α α, β β α β, So, the proof is completed Similarly the other euatio ca be obtaied by the euatio Q ropositio 7 If, Q, are ell, ell-lucas ad Modified ell umbers, the for all itegers, we k Q + k roof: Cosiderig propositio, we ca write ( Q k k k+ k k k k ( + ( ( Thus, the proof is completed ropositio If, are th ell ad Modified ell umbers, the for all itegers,, we ad Q, + k k ( + ( Q+ 6

5 SAÜ Fe Bilimleri Dergisi, Cilt, Sayı, s-5, 00 O Some Relatioships Amog ell, ell-lucas ad Modified ell Seueces roof: I [], author give followig relatio betwee the ell ad ell-lucas umbers + ( Q ( + If we take istead of i last euatio, the we ( Q Thus, we obtai that Q k k + k k ( ( which is desired Similarly, the other euatio ca be obtaied ropositio 9 Let α, β be the root of x x 0 The ad α +, β roof: We will prove the theorem by iductio method o By the defiitios of ell ad Modified ell umbers, we α So, the proof is completed Also, we ca write β Thus, we get α +, β which is desired RERERENCES [] J Ercolao, Matrix geerator of ell seuece, Fiboacci Quart 7, (979, 7-77 [] A F Horadam, ell idetities, Fiboacci Quart 9,3 (97, 5-5 [3] R Melham, Sums Ivolvig Fiboacci ad ell Numbers, ortugaliae Math, 56(3, , 999 [] AF Horadam, Applicatios of Modified ell Numbers to Represetatios, Ulam Quarterly, 3(99, 3-53 [5] D Zielberger, The method of creative telescopig, J Symb, Comp, Vol (99, 95-0 [6] E Kılıc ad D Tascı,,The Liear Algebra of The ell Matrix, Bull Soc Mat Mexicaa,, (005, 63-7 [7] T Koshy, Fiboacci ad Lucas Numbers with Applicatios, Wiley-, (00 α + We suppose that the claim is true for Now, we will show that the claim is true for Usig by our assumptio, we ca write α ( ( + α α I [], author gave a relatioships such that +, + Therefore, we obtai that 5