THE ADJACENCY-PELL-HURWITZ NUMBERS. Josh Hiller Department of Mathematics and Computer Science, Adelpi University, New York

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1 #A8 INTEGERS 8 (8) THE ADJACENCY-PELL-HURWITZ NUMBERS Josh Hller Departent of Matheatcs and Coputer Scence Adelp Unversty New York johller@adelphedu Yeş Aküzü Faculty of Scence and Letters Kafkas Unversty Turkey Öür Devec Faculty of Scence and Letters Kafkas Unversty Turkey Receved: 9// Revsed: 8//8 Accepted: //8 Publshed: //8 Abstract In ths paper we defne the k-adjacency-pell-hurwtz nubers by usng the Hurwtz atrx of order whch s obtaned by the ad of the characterstc polynoal of the adjacency-pell sequence Frstly we gve relatonshps between the k-adjacency- Pell-Hurwtz nubers and the generatng atrces for these sequences Further we obtan the Bnet forula for the ( )-adjacency-pell-hurwtz nubers Also we derve relatonshps between the k-adjacency-pell-hurwtz nubers and peranents and deternants of certan atrces Fnally we gve the cobnatoral and exponental representatons of the k-adjacency-pell-hurwtz nubers Introducton It s well-known that the Pell sequence s defned by the followng equaton: P n+ P n + P n for n > where P P The adjacency-type sequence s defned n [] by an n-order recurrence equaton: x n n+k xn n n++k + xn k for k where x n x n n n+ x n n and n xn n n+ xn n n+

2 INTEGERS: 8 (8) Karaduan and Devec defned the adjacency-pell sequence as follows: a n (n + k) a n (n n + k + ) + a n (k) for the ntegers k and n wth ntal constants a n () a n (n ) and a n (n) [] Consder the k-step recurrence sequence: a n+k c a n + c a n+ + + c k a n+k where c c c k are real constants Earler Kalan [8] derved a nuber of closed-for forulas for soe generalzed sequences va the copanon atrx ethod as follows: If the copanon atrx A s defned by then for n > Consder f A [a j ] k k A n c c c c k c k a a a k a n a n+ a n+k a real polynoal of degree n gven by f (x) a x n + a x n + + a n x + a n

3 INTEGERS: 8 (8) Hurwtz [] ntroduced the atrx H n [h j ] n n assocated to f as follows: a a a a a a a a a a H n a an a an a n a n a n a n a n a n a n Several suthors have used hoogeneous lnear recurrance relatons to deduce scellaneous propertes for a plethora of sequences; see for exaple [ 9 8 9] In partcular Devec and Shannon defned the adjacencytype nubers and exaned ther structural propertes [] The adjacency-pell nubers ther scellaneous propertes and applcatons n groups were studed by Devec and Karaduan n [] In the present paper we defne the k-adjacency- Pell-Hurwtz nubers by a recurrence relatons of order ( ) and gve ther generatng atrces Bnet forulas peranental deternantal cobnatoral exponental representatons and we derve a forula for the sus of the k-adjacency-pell-hurwtz nubers The Man Results For and n t s clear that the characterstc polynoal of the adjacency- Pell sequence s p (x) x x () Then by () we see that the Hurwtz atrx H [h j ] assocated to a polynoal p s 8 >< [h j ] >: f k and j k + for apple k apple f k and j k + for apple k apple f k and j k for apple k apple otherwse We defne the k-adjacency-pell-hurwtz nubers by usng the Hurwtz atrx H as shown: x (k) +u x(k) k++u + x(k) k +u ()

4 INTEGERS: 8 (8) for the ntegers u and apple k apple wth ntal constants x (k) x (k) and x(k) Here x(k) s the th ter of the kth sequence accordng to the constant By () we ay wrte h M ( ) h M () ( ) j () j ( + ) th #

5 INTEGERS: 8 (8) for apple apple and h ( ) M ( ) j We call atrx M ( ) the -adjacency-pell-hurwtz atrx of sze By an nductve arguent on we obtan M () for x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + x () + M ( ) E + + +

6 INTEGERS: 8 (8) for and apple apple where E s the followng ( + ) atrx: E and M ( ) x ( ) ( ) + x x ( ) ( ) + x x ( ) ( ) + x x ( ) + x x ( ) + x x ( ) + x + + x + x + x ( ) + x ( ) + x ( ) + x ( ) ( ) ( ) ( ) 8+ x + x + x + ( ) ( ) ( ) ( ) 8+ x + x + x + ( ) ( ) ( ) ( ) 8+ x + x + x + ( ) ( ) ( ) ( ) + x x + x + ( ) ( ) ( ) ( ) + x x x + ( ) ( ) ( ) ( ) + x x x for Let and let S (k) t t P ntroduce atrx H (k ) by H (k ) x (k) such that apple k apple We M (k) for apple k apple Note that H (k ) s a square atrx of sze ( + )

7 INTEGERS: 8 (8) ( + ) and t can be shown by nducton that: S () + S () + S () + (H ( )) S () + M () for S ( ) + S ( ) + (H ( )) S ( ) + M ( ) for and apple apple and (H ( )) S ( ) + S ( ) + S ( ) + S ( ) + S ( ) M ( ) ( ) Lea The equaton x + x nteger does not have ultple roots for any Proof Let q (x) x + x and suppose v s a ultple root of q (x) Snce q () t follows that v Then the hypotheses q (v) and q (v) ply v and v respectvely It follows that v > and v < nequaltes that cannot hold sultaneously for Ths s a contradcton resultng fro our assupton that v s a ultple root whch concludes the proof of the lea

8 INTEGERS: 8 (8) 8 ( ) Let q (v) be the characterstc polynoal of atrx M Then q (v) v + v ( ) a clear fact because M s a copanon atrx ( ) Let v v v be the egenvalues of M By Lea we know that ( ) these are dstnct nubers Let V be the followng Vanderonde atrx: ( ) V (v ) (v ) (v ) (v ) (v ) (v ) v v v ( ) Denote by V ( j) the atrx obtaned fro V colun by ( ) C ( j) v + v + v + We can gve the generalzed Bnet forula for the ( nubers wth the followng theore h ( ) Theore For the atrx M ( j ( ) j det V ( ) ( ) det V ( ) by replacng the jth )-adjacency-pell-hurwtz ) for ( j) () Proof Consder the nteger to be fxed Snce v v v are dstnct the ( ) ( ) ( ) ( ) atrx M s dagonalzable Then M V V D where ( ) D (v v v ) Snce det V we can wrte ( ) V M ( ) ( ) V D ( ) Then the atrx M s slar to D and so ( ) M V ( ) ( ) V (D ) We can now easly establsh the followng lnear syste of equatons: 8 ( ) (v ) ( ) + (v ) ( ) + + (v ) n+ >< ( ) (v ) ( ) + (v ) ( ) + + (v ) n+ >: ( ) ( ) ( ) (v ) + (v ) + + (v ) n+ The nubers n forula () are solutons of the last lnear syste

9 INTEGERS: 8 (8) 9 Theore gves edately: Corollary Let x ( ) be the th eleent of the ( )-adjacency-pell- Hurwtz sequence then x ( ) det V ( ) ( ) det V ( ) ( ) det V ( ) ( ) det V ( ) det V ( ) ( ) det V Now we consder the peranental representatons of the k-adjacency-pell-hurwtz nubers Defnton Let M [ j ] be u v real atrx and let r r r u and c c c v be respectvely the row and colun vectors of M If r contans exactly two non-zero entres then M s contractble on row Slarly M s contractble on colun provded c contans exactly two non-zero entres Let x x x u be row vectors of the atrx M and let M be contractble n the th colun wth j and j Then the (u ) (v ) atrx M j: obtaned fro M by replacng the th row wth x j + j x and deletng the j th row and the th colun s called the contracton n the th colun relatve to the th row and the j th row The peranent of a u-square atrx A [a j ] s defned by per(a) X S u u Y a () where the suaton extends over all perutatons of the syetrc group S u In [] Bruald and Gbson showed that per (A) per (B) f A s a real atrx of order u > and B s a contracton of A h Let n and let X (k ( n)) x nk j super-dagonal atrces defned usng the followng cases: and j + k for apple apple n k apple k apple be the n n j + k + for apple apple n k and j for apple apple n otherwse x nk j 8 >< f case () apples f case () apples >: f case() apples

10 INTEGERS: 8 (8) where s as n the defnton of the k-adjacency-pell-hurwtz nubers Then we have the followng theore Theore For apple k apple we have per(x (k ( n))) x (k) +n Proof Consder the atrx X ( ( n)) We wll use nducton on n Assue the equaton holds for n Then we ust show that the equaton per(x ( ( n))) x () +n holds for n + If we expand per(x ( ( n))) by the Laplace expanson of peranent accordng to the frst row then we obtan per(x ( ( n + ))) per(x ( ( n))) + per(x ( ( n ))) Snce per(x ( ( n ))) x () +n we obtan per(x ( ( n + ))) x () +n + x() +n x() +n+ The proofs for apple k apple are slar to the above and are otted Let n Now we defne the n n atrces Y (k ( n)) h y nk j apple k apple usng the followng cases: and j + k for apple apple n k j + k for apple apple n k and j for apple apple n otherwse Wth these three cases n nd we now defne the desred atrx: 8 f case () apples >< y nk j f case () apples >: f case() apples where s as n the defnton of the k-adjacency-pell-hurwtz nubers In the next theore we obtan another peranental representaton Theore For apple k apple we have per(y (k ( n))) x (k) k +n Proof Consder atrces Y ( ( n)) h y n j apple apple We wll use nducton on n Suppose that the equaton holds for n Then we ust show

11 INTEGERS: 8 (8) that the equaton holds for n + If we expand the pery ( ( n)) by the Laplace expanson of peranent accordng to the frst row then we obtan per(y ( ( n + ))) per(y ( ( n + )))+per(y ( ( n ))) ) ) +n + x( +n x( +n for apple apple Thus the concluson s obtaned The proofs for the atrces Y ( ( n)) and Y ( ( n)) are slar We now consder the sus of the k-adjacency-pell-hurwtz nubers by usng ther peranental representatons Let n > and suppose that Z ( ( n)) Z ( ( n)) ( apple apple ) and Z ( ( n)) are the n n atrces defned by and (n ) th # Z ( ( n)) Y ( ( n )) (n ) th # Z ( ( n)) Y ( ( n )) () () (n ) th # Z ( ( n)) Y ( ( n )) Then we have the followng theore ()

12 INTEGERS: 8 (8) Theore For apple k apple ()() and () Then and per(z ( ( n))) per(z ( ( n))) let the n n atrces Z (k ( n)) be as n n+x " n+ X " nx per(z ( ( n))) x () " " ( apple apple ) x ( " ) " Proof Consder the atrces Z( ( n)) Expandng per(z( ( n))) wth respect to the frst row we have per(z ( ( n))) per(z ( ( n )))+per(y ( ( n ))) By Theore and the nductve arguent on n we easly coplete the proof The proofs for apple k apple are slar to the above and are otted A atrx M s called convertble f there s an n n ( )-atrx K such that perm det (M K) where M K denotes the Hadaard product of M and K Let n > and let W be the n n atrx defned by W and It s easy to see that per(x (k ( n)) det (X (k ( n)) W )) per(y (k ( n)) det (Y (k ( n)) W )) per(z (k ( n)) det (Z (k ( n)) W )) for n > and apple k apple Consder the n n atrx c c c n C C (c c c n )

13 INTEGERS: 8 (8) For detaled nforaton about the copanon atrx see [ p9] and [ p8] Theore (Chen and Louck [])The ( j) entry c ( ) j (c c c n ) n the atrx C (c c c n ) s gven by the followng forula: c ( ) j (c c c n ) X (t t t n) t j + t j+ + + t n t + t + + t n t + + t n t t n c t ctn n () where the suaton s over nonnegatve ntegers satsfyng t + t + + nt n t + j + +t n t t n (t+ +tn)! t!t n! s a ultnoal coe cent and the coe cents n () are defned to be f j Now we gve the cobnatoral representatons for the ( Hurwtz sequence by the followng Corollary )-adjacency-pell- Corollary Let x ( ) be the th eleent of the ( )-adjacency-pell- Hurwtz sequence such that and Then X x ( ) t + t + t t + + t ( ) t t + t + + t t t (t t t ) X t + t t + + t ( ) t t + t + + t t t (t t t ) X t t + + t ( ) t t + t + + t t t (t t t ) where the suaton s over nonnegatve ntegers satsfyng t +t + +() t Proof In Theore f we choose n and j such that apple j apple then the proof follows fro () Let Now we gve the generatng functon of the k-adjacency-pell-hurwtz nubers g (k) (y) x (k) Then y k + x(k) + y + x(k) + y + + x (k) +u g (k) (y) x (k) yk + x (k) + yk + x (k) + yk+ yu + x (k) +u yu x (k) +u yk +u + x (k) +u yk+u + and y k+ g (k) (y) x (k) yk+ + x (k) + yk+ + x (k) + yk+

14 INTEGERS: 8 (8) Thus we have + + x (k) +u yk++u + x (k) +u yk++u+ + g (k) (y) + y k g (k) (y) y k+ g (k) (y) x (k) y By the defnton of the k-adjacency-pell-hurwtz nubers we obtan g (k) (y) y + y k y k+ where apple k apple and y k+ y k < Now we gve an exponental representaton for the k-adjacency-pell-hurwtz nubers Theore 8 For apple k apple and y k+ y k < the k-adjacency- Pell-Hurwtz nubers have the followng exponental representaton: X g (k) (y) y y k! exp y Proof Snce and we have ln g (k) (y) ln y + y k y k+ ln y ln + y k y k+ ln + y k y k+ ln y k+ y k Therefore we obtan [ y k y + yk y X ln g (k) (y) ln y ln g (x) y X + + y k y k X y k y k y + ]! y y y The last forula ples the one n the text of the theore thus concludng the proof

15 INTEGERS: 8 (8) Acknowledgeent Ths work was supported by the Cosson for the Scentfc Research Projects of Kafkas Unversty Project nuber -FM- The authors would lke to thank the anonyous referee for a careful readng and any helpful coents that proved the fnal verson of ths anuscrpt References [] R Bruald and P Gbson Convex polyhedra of doubly stochastc atrces I: applcatons of peranent functon J Cobn Theory (9) 9- [] W Chen and J Louck J The cobnatoral power of the copanon atrx Lnear Algebra Appl (99) -8 [] O Devec and A Shannon On the adjacency-type sequences Int J Adv Math () - [] O Devec Y Akuzu and E Karaduan The Pell-Padovan p-sequences and Its Applcatons Utl Math 98 () - [] O Devec and E Karaduan On The Adjacency-Pell Nubers Utl Math n press [] N Gogn and A Myllar The Fbonacc-Padovan sequence and MacWllas transfor atrces Progra Coput Softw publshed n Prograrovane () () -9 [] A Hurwtz Ueber de Bedngungen unter welchen ene glechung nur Wurzeln t negatve reellen telen bestzt Matheatsche Annalen (89) -8 [8] D Kalan Generalzed Fbonacc nubers by atrx ethods Fbonacc Quart () (98) - [9] E Klc The generalzed Pell (p)-nubers and ther Bnet forulas cobnatoral representatons sus Chaos Soltons Fractals () (9) - [] E Klc and A Stakhov On the Fbonacc and Lucas p-nubers ther sus fales of bpartte graphs and peranents of certan atrces Chaos Soltons Fractals (9) - [] P Lancaster and M Tsenetsky The Theory of Matrces Acadec 98 [] R Ldl and H Nederreter Introducton to Fnte Felds and Ther Applcatons Cabrdge UP Cabrdge 99 [] N Ozgur On the sequences related to Fbonacc and Lucas nubers J Korean Math Soc () () - [] A Shannon P Anderson and A Horada Propertes of cordonner Perrn and Van der Laan nubers Internat J Math Ed Sc Tech () () 8-8 [] A Shannon and L Bernsten The Jacob-Perron algorth and the algebra of recursve sequences Bull Australan Math Soc 8 (9) - [] A Stakhov and B Rozn Theory of Bnet forulas for Fbonacc and Lucas p-nubers Chaos Soltons Fractals () - [] D Tascı and M Frengz Incoplete Fbonacc and Lucas p nubers Math Coput Modellng (9) () -

16 INTEGERS: 8 (8) [8] N Tuglu E Kocer and A Stakhov Bvarate Fbonacc lke p-polynoals Appl Math and Copt () 9- [9] F Ylaz and D Bozkurt The generalzed order-k Jacobsthal nubers Int J Contep Math Sc () (9) 8-9