Hacettepe Joural of Mathematic ad Statitic Volume 4 4 03, 387 393 AN APPLICATION OF HYPERHARMONIC NUMBERS IN MATRICES Mutafa Bahşi ad Süleyma Solak Received 9 : 06 : 0 : Accepted 8 : 0 : 03 Abtract I thi tudy, firtly we defied a k matrix, G r, whoe etrie coit of hyperharmoic umber The we obtaied relatio betwee Pacal matrice ad G r Fially we calculated the determiat of G r, Keyword: Pacal Matrix; Hyperharmoic umber; Determiat 000 AMS Claificatio: B99; C0; 5A3 Itroductio For > 0 ad i k, let defie the order-k equece be a followig: k g i c jg j i j with iitial value g k, i g k, i, g0, i where c j j k are cotat coefficiet, g i i the th term of ith equece Let the k k matrix be a followig: g g g k g g g k G g k+ g k+ g k+ k There have bee may paper related to the equece a i [,, 3, 4, 5] I [], Kalma obtaied a umber of cloed-form formula for the geeralized equece by matrix method Akaray Uiverity, Educatio Faculty, Akaray TURKEY Email: M Bahşi mhvbahi@yahoocom N E Uiverty, A K Educatio Faculty, Koya TURKEY Email: S Solak olak4@yahoocom Correpodig author
388 M Bahşi ad S Solak I [], Er defied the order-k Fiboacci umber a a equece which atifie the recurrece with the boudary coditio for k 0 {, if i g i 0, otherwie Whe k ad c j j k, thi reduce to the well-kow covetioal Fiboacci umber Alo, Er howed that [ ] g i + g i g k+ i T [ ] C g i g i g k+ i T, where C c c c 3 c k c k 0 0 0 0 0 0 0 0 The he obtaied G + CG, 0 0 0 0 0 0 0 0 0 where G i k k matrix a i I [3], Karaduma howed that ad G C det G {, if k i eve, if k i odd for c j j k I [4], Taci ad Kilic gave a ew geeralizatio of the Luca umber i matrice Alo, they preeted a relatio betwee the geeralized order-k Luca umber ad Fiboacci umber I [5], Fu ad Zhou obtaied ome ew reult o matrice related to Fiboacci ad Luca umber The th hyperharmoic umber of order r, H r, defied a: for, r 3 H r where H 0 H r k k From the defiitio of Hr, we have H r, ad H k H k where H i th ordiary harmoic umber Alo, hyperharmoic umber have the recurrece relatio a follow: H r H r + H r I [6], Coway ad Guy gave a equality a follow: H r + r r H +r H r ad i [7], Bejami ad et all gave 4 H r + r r
A applicatio of hyperharmoic umber i matrice 389 Let the k matrix be a followig: 5 G r H r H r H r+ H r+k H r+ H r+k H r H r+ H r+k where H r i th hyperharmoic umber of order r defied a i 3 I Sectio, we derive the relatio betwee Pacal matrice ad G r Alo, we calculate the determiat of G, r Now we give ome prelimiarie related to our tudy The k Pacal ad lower triagular Pacal matrice are repectively defied a 6 P p ij 7 P L q ij i + j j { i j, if i j, 0, otherwie For example, the matrice P ad P L of order 5 are 0 0 0 0 3 4 5 P 3 6 0 5 4 0 0 35, PL 0 0 0 0 0 3 3 0 5 5 35 70 4 6 4 The matrice P ad P L i 6 ad 7 have the followig propertie [8, 9] : P P LP T L, where P T L i trapoe of P L DetP 3 P L diag [,,,, ] P Ldiag [,,,, ] 4 P diag [,,,, ] P T L P L diag [,,,, ] Let the matrice H ad A be a 8 H 3 0 0 0 0 0 0 ad 0 9 A 0 0
390 M Bahşi ad S Solak The from the priciple of mathematical iductio o r, we have { j i+r 0 A r r, if i j B r b ij 0, otherwie Alo, the determiat of A ad H have the form: ad det A det H The Mai Reult {, if 0, mod 4, if, 3 mod 4 Lemma Let the k,, k matrice G r, H, P be a i 5, 8 ad 6, repectively The, G HP Proof From matrix multiplicatio, we have + HP 3 + + 4 + k k + k k k + k From 4 ad ice H r, H H H k H H H k HP Thu, the proof i completed H H H k G Lemma Let the k matrice G r, P ad matrice H, A be a i 4, 6, 8 ad 9, repectively The, G r+ A r HP Proof From matrix multiplicatio ad 3, we have [ ] T [ H r+ H r+ H r+ A Geeralizig, we derive G r+ AG r H r By uig the priciple of mathematical iductio, we write G r+ A r G H r H r ] T
A applicatio of hyperharmoic umber i matrice 39 From Lemma, the Eq i rewritte a G r+ A r HP 3 Corollary Let the k matrice G r, P ad matrice H, Br i 5, 6, 8 ad 0, repectively The, G r+ B rhp For example, takig 4, k 3 ad r 5 i Corollary 3, we have G 6 4,3 75 4 73 3 3 07 9 6 5 69 3 7 5 5 35 0 5 5 0 0 5 0 0 0 4 3 0 3 0 0 0 0 0 3 3 6 4 0 B5HP be a 4 Theorem Let the th hyperharmoic umber of order r, H r, be a i 3 The, we have H r+ t where r ad 0 Proof Let the matrice G r where G r+ we have G r+ g H r+, b j g + r t r H t ad Br be a i 5 ad 0, repectively The j + r ad q j H j+ r g ij, B r b ij ad G qij From Lemma ad Corollary 3, B rg The j j t b jq j j + r r + r t r H j+ H t Sice g H r+, the proof i completed Takig r i Theorem 4, we ca write H r + r t H r t r t
39 M Bahşi ad S Solak Alo, takig r + i Theorem 4, we have 3 H H + ad 4 H H +0 H t t t H t t + + H+ t t Therefore, from 3 ad 4, for um of the firt ordiary harmoic umber, we obtai H t + H + t 5 Theorem Let the matrix G r be a i 5 The det G, r {, if 0, mod 4, if, 3 mod 4 Proof From Lemma, for k, we write G r, A r HP The det G, r [det A] r det H det P Sice det H {, if 0, mod 4, if, 3 mod 4, det A ad det P, we have { det G, r, if 0, mod 4, if, 3 mod 4 ad Takig i Theorem 5, we have det G r H r H r+, H r H r+ H r+ H r where H r Sice H 3, thu we have H r + r
A applicatio of hyperharmoic umber i matrice 393 Referece [] Kalma D Geeralized Fiboacci umber by matrix method, Fiboacci Quarterly 0, 73 76, 98 [] Er MC Sum of Fiboacci umber by matrix method, Fiboacci Quarterly 3, 04 07, 984 [3] Karaduma E A applicatio of Fiboacci umber i matrice, Applied Mathematic ad Computatio 47, 903 908, 004 [4] Taci D ad Kilic E O the order-k geeralized Luca umber, Applied Mathematic ad Computatio 55, 637 64, 004 [5] Fu X ad Zhou X O matrice related with Fiboacci ad Luca umber, Applied Mathematic ad Computatio 00, 96 00, 008 [6] Coway, JH ad Guy, RK The Book of Number, Spriger-Verlag, New York, 996 [7] Bejami, AT, Gaebler, D ad Gaebler, R A combiatorial approach to hyperharmoic umber, Iteger: Electro J Combi Number Theory 3, 9, 003 [8] El-Mikkawy, MEA O olvig liear ytem of the Pacal type, Applied Mathematic ad Computatio 36, 95 0, 003 [9] Lv, X-G, Huag, T-Z ad Re, Z-G A ew algorithm for liear ytem of the Pacal type, Joural of Computatioal ad Applied Mathematic 5, 309 35, 009