A generalization of Fibonacci and Lucas matrices

Transcription

1 Discrete Applied Mathematics 56 28) wwwelseviercom/locate/dam A geeralizatio of Fioacci ad Lucas matrices Predrag Staimirović, Jovaa Nikolov, Iva Staimirović Uiversity of Niš, Departmet of Mathematics, Faculty of Sciece, Višegradska 33, 8 Niš, Seria Received 27 Feruary 27; received i revised form 2 Septemer 27; accepted 29 Septemer 27 Availale olie 2 Feruary 28 Astract We defie the matrix U a,,s) of type s, whose elemets are defied y the geeral secod-order o-degeerated sequece ad itroduce the otio of the geeralized Fioacci matrix F a,,s), whose ozero elemets are geeralized Fioacci umers We oserve two regular cases of these matrices s ad s ) Geeralized Fioacci matrices i certai cases give the usual Fioacci matrix ad the Lucas matrix Iverse of the matrix U a,,s) is derived I partial case we get the iverse of the geeralized Fioacci matrix F a,,) ad later kow results from Gwag-Yeo Lee, Ji-Soo Kim, Sag-Gu Lee, Factorizatios ad eigevalues of Fioaci ad symmetric Fioaci matrices, Fioacci Quart 4 22) 23 2; P Stǎicǎ, Cholesky factorizatios of matrices associated with r-order recurret sequeces, Electro J Comi Numer Theory 5 2) 25) #A6] ad Z Zhag, Y Zhag, The Lucas matrix ad some comiatorial idetities, Idia J Pure Appl Math i press)] Correlatios etwee the matrices U a,,s), F a,,s) ad the geeralized Pascal matrices are cosidered I the case a, we get kow result for Fioacci matrices Gwag-Yeo Lee, Ji-Soo Kim, Seog-Hoo Cho, Some comiatorial idetities via Fioacci umers, Discrete Appl Math 3 23) ] Aalogous result for Lucas matrices, origiated i Z Zhag, Y Zhag, The Lucas matrix ad some comiatorial idetities, Idia J Pure Appl Math i press)], ca e derived i the partial case a 2, Some comiatorial idetities ivolvig geeralized Fioacci umers are derived c 27 Elsevier BV All rights reserved Keywords: Fioacci umer; Lucas umer; Fioacci matrix; Lucas matrix Itroductio The Fioacci umers {F } are the terms of the sequece,,, 2, 3, 5, where each term is the sum of the two precedig terms, ad we get thigs started with ad as F ad F You caot go very far i the lore of Fioacci umers without ecouterig the compaio sequece of Lucas umers {L }, which follows the same recursive patter as the Fioacci umers, ut egis with L 2 ad L The sequece of Lucas umers is therefore 2,, 3, 4, 7, 3] We also oserve so-called geeralized Fioacci umers, {F a,) }, which satisfy the same recursive formula F a,) 2 Fa,) Fa,),,,, ut startig with aritrary iitial values F a,) a ad F a,), see for example 9,6,2], ], Chapter 7)) Correspodig author Fax: addresses: pecko@pmfiacyu P Staimirović), JovaaNikolov@gmailcom J Nikolov), IvaStaimirovic@gmailcom I Staimirović) 66-28X/$ - see frot matter c 27 Elsevier BV All rights reserved doi:6/dam27928

2 P Staimirović et al / Discrete Applied Mathematics 56 28) The Fioacci matrix F f ],, ) is defied y 7]: f { Fi, i,, i < ) The iverse ad Cholesky factorizatio of the Fioacci matrix are give i 7] The relatios etwee the Pascal matrix ad the Fioacci matrix are studied i 8] As a aalogy of the Fioacci matrix, the Lucas matrix L l ],, ) is defied i 6]: l { Li, i,, i < 2) I the paper ] the author ivestigated the iverse ad Cholesky factorizatio of the matrix U with etries u { Ui, i,, i <, 3) where U is the o-degeerated secod order sequece U AU BU, δ A 2 4B real, ad where A, B, U are itegers ad U ie A B) I ] the author also geeralized these results to r-order recurret sequece satisfyig U U U 2r, U aritrary Results otaied i ] iclude kow facts aout the Fioacci matrix 7,8] i the case U, A B But, results aout the Lucas matrices from 6] are ot icluded Lucas sequece is geerated y the associated sequece V which satisfy V 2, V a Our goal i this paper is to geeralize all results aout the Fioacci ad Lucas matrices The purpose of this paper is to demostrate that kow properties of Fioacci, Lucas matrices ad the matrices defied i ] are valid for a more geeral class of matrices, itroduced i Sectio 2 Throughout the paper we adopt the followig two covetios: ad ) k for k >, eve i the case k By raka) we deote the rak of matrix A The paper is orgaized as follows I Sectio 2 we defie the matrix U a,,s) umers U a,) satisfyig the geeral secod order o-degeerated recurrece formula U a,) of type s, whose etries are a,) AU BU a,), δ A 2 4B real, ad iitial coditios U a,) a, U a,) I the case A B we itroduce the geeralized Fioacci matrix F a,,s) of type s, whose ozero elemets are geeralized Fioacci umers F a,) Oly two cases geeratig regular matrices are s ad s Geeralized Fioacci matrices reduce to kow defiitio of the usual Fioacci matrix i the cases s, a, ad s, a, I the case a 2,, s we otai the matrix whose ozero etries are Lucas umers, ad arraged as i the Fioacci matrix This matrix is called the Lucas matrix 6] At this momet we cosider the matrices U a,,) ad F a,,) Iverses of the geeralized Fioacci matrix ad for the matrix U a,,) are derived I the partial case a, we get kow result aout the iversio of the usual Fioacci matrix from 7] Similarly, i the case a 2, we otai the iverse of the Lucas matrix, origiated i 6] Moreover, i Sectio 2 we cosider the matrix U a,,) defied y meas of the geeral o-degeerated secod-order recurret sequece, ad geeralize Propositio 2 from ] Various correlatios etwee the matrix U a,,s) ad the Pascal matrix of the first ad the secod kid are cosidered i Sectio 3 Correspodig results for the geeralized Fioacci matrix F a,,) are give as corollaries Partial case a, produces kow result from 8] I the case a 2, we derive aalogous results for Lucas matrices, ivestigated i 6] I Sectio 4 we get some comiatorial idetities ivolvig geeralized Fioacci umers ad iomial coefficiets 2 Geeralized Fioacci matrix ad its iverse By F a,) we deote the -th geeralized Fioacci umer, geerated y the Fioacci recursive formula ad y the iitial values F a,) a, F a,) Notios of Fioacci ad Lucas matrix are geeralized i the followig defiitio Defiitio 2 Let F a,) e the -th geeralized Fioacci umer, where the startig memers of the Fioacci array are F a,) a ad F a,), ad where a, C The geeralized Fioacci matrix of type s ad of the order

3 268 P Staimirović et al / Discrete Applied Mathematics 56 28) , deoted y F a,,s) f a,,s) { f a,,s) ], is defied y F a,) i, i s,,, 2), i s < It is clear that the iteger s meas the shift of o-zero elemets with respect to mai diagoal We also defie a geeralizatio U a,,s) u a,,s) ] of the matrix F a,,s) ad the matrix U from 3) Defiitio 22 The matrix U a,,s) u a,,s) { U a,) i, i s,, i s <, u a,,s) ] is defied y 22) where the secod order recurret sequece U a,) satisfies the followig coditios: U a,) AU a,) a,) BU 2, U a,) a, U a,), A 2 4B > 23) Remark 2 a) Geeralized Fioacci matrices F,,) ad F,,) are oth idetical to the usual Fioacci matrix defied i ) ) The geeralized Fioacci matrix F 2,,) correspods to Lucas matrix, defied i 2) c) The matrix U,,) reduces to the matrix U defied i 3) Example 2 The 6 6 geeralized Fioacci matrix of type is equal to a F a,,) a 2 a 6 2a 3 a 2 a 3a 5 2a 3 a 2 a 5a 8 3a 5 2a 3 a 2 a The 6 6 geeralized Fioacci matrix of type is defied y a a a F a,,) a 2 a a 6 2a 3 a 2 a a 3a 5 2a 3 a 2 a a 5a 8 3a 5 2a 3 a 2 a The matrix U a,,) 4 is equal to U a,,) 4 A ab B AA ab) A ab BA ab) AB AA ab)) B AA ab) A ab Propositio 2 The odegeerated secod-order recurret sequece U a,), defied i 23), satisfies the followig geeralizatio of the Biet s Fioacci umer formula U a,) c α c 2 β, 24)

4 P Staimirović et al / Discrete Applied Mathematics 56 28) where c aa2 4B) 2 a A) A 2 4B 2A 2, 25) 4B) c 2 aa2 4B) 2 a A) A 2 4B 2A 2, 26) 4B) α A A 2 4B 2, β A A 2 4B 27) 2 I the case s > it is ecessary to use geeralized Fioacci umers F a,) ad the umers U a,) with egative idices Recurret defiitio of the geeralized Fioacci umers ca e expaded for egative idices usig 24) 27), similarly as for the Fioacci umers i ] Lemma 2 The followig idetity is valid for the secod order o-degeerated recurret sequece U a,) ad for two aritrary itegers satisfyig i 2: a 2 B aa 2 ) Proof By usig 27) we otai αβ B, α β A, α β satisfyig ) k ak 2 B k U a,) k ik ab 2 U a,) i B U a,) i 28) By applyig 24) ad simple trasformatios, we otai the followig: Usig we get a 2 B aa 2 ) a 2 B aa 2 ) a2 B aa 2 3 ab α a 2 B aa 2 ) ) k ak 2 B k U k ) k 2 ab ) i α a2 B aa 2 3 A 2 4B 29) a,) ik ) k ak 2 Bk k c α ik c 2 β ik) ab ) k 2 Bα i c ab ) ) k 2 Bβ i c 2 α β ab α ) k ak 2 B k U k ab α c ab α With cosideratio of 29), we have a 2 B aa 2 ), )i ) k ak 2 B k U k a,) ik ab ) k 2 β α i ab β c 2 ab β a,) ik )i ab β ab β β i ) B ) i

5 26 P Staimirović et al / Discrete Applied Mathematics 56 28) a2 B aa 2 3 ab β) α i α ab )i ) c ab α) β i β ab )i ) c 2 a A a2 B 2 c ab αi α ab ) i Bα i B ab ) ) i ab c 2 βi β ab ) i Bβ i B ab ) )] i By groupig similar memers, usig c c 2 a, c α c 2 β U a,) ad usig 24) ad 29), oe ca verify the followig: a 2 B aa 2 ) ) k ak 2 B k U k ab c α i c 2 β i ) c α c 2 β) a,) ik ab B c α i c 2 β i ) Bc c 2 ) ab ab 2 U a,) i U a,) ab ) i B U a,) i ab ) i ) i ] ab ) i The proof ca e completed y usig U a,) I the partial case A B we otai the followig result for the geeralized Fioacci umers Corollary 2 For the geeralized Fioacci umers F a,), ad for two aritrary itegers satisfyig i 2 the followig is valid: a 2 a 2 ) ) k ak 2 k Fa,) ik a 2 Fa,) i I the case a 2,, from the previous corollary we get kow result from 6] Corollary 22 For the Lucas umers ad each i 2 the followig is valid: 5 ) k 2 k 2 L ik 2L i L i Fa,) i 2) Theorem 2 The iverse U a,,) u a,,) u a,,) ] of the matrix U a,,) u a,,) ] ) is equal to ) i a2 B aa 2 i a i 2 B i, i 2, ab A 2, i,, i,, i < 2) Proof Let k u a,,) i,k u a,,) k, c Oviously c for i < I the case i oe ca verify the followig: c i,i u a,,) i,i u a,,) i,i

6 P Staimirović et al / Discrete Applied Mathematics 56 28) I the case i we otai c, u a,,), u a,,), u a,,), u a,,), A Ba) ) ab A 2 For i 2, y applyig results of Lemma 2 ad 23) ad 2) we otai c k u a,,) i,k u a,,) k, u a,,) u a,,), u a,,) u a,,) U a,) i U a,) i ab A 2 U a,), u a,,) i,k u a,,) k, i a 2 B aa 2 ) ab A 2 U a,) i ab 2 U a,) i B U a,) i a,) a,) U i AU i BU a,) i ) Therefore, we verify U a,,) U a,,) U a,,) U a,,) I ) k ak 2 B k U k a,) ik I, where I is idetity matrix I a similar way oe ca verify Example 22 The iverse of the matrix U a,,) 4 is equal to A ab 2 B Ba 2 Aa 2) A ab 3 2 ab 2 Ba 2 Aa 2) B Ba 2 Aa 2) A ab Remark 22 Propositio 2 from ] ca e derived y placig a i 2) I the partial case A B from Theorem 2 we otai the iverse of the geeralized Fioacci matrix of type Corollary 23 Let F a,,) F a,,), deoted y F a,,) f a,,) ], is equal to f a,,) ],, e geeralized Fioacci matrix of type The iverse of ) i a2 a 2 i a i 2, i 2, f a,,) a 2, i,, i,, i < 22)

7 262 P Staimirović et al / Discrete Applied Mathematics 56 28) Example 23 The iverse of the geeralized Fioacci matrix F a,,) 6 is equal to a 2 a 2 a 2 3 a 2 aa2 a 2 ) a 2 a a 2 a 2 a 2 a 2 ) 5 aa2 a 2 ) a 2 a a3 a 2 a 2 ) a 2 a 2 a 2 ) 6 5 aa2 a 2 ) a 2 a I the case a 2, we get the iverse Lucas matrix, derived i 6] Corollary 24 The iverse of the Lucas matrix L 5) i 2 i 2, i 2, l i, 3, i,, i,, otherwise l i, ],, ) is equal to a 2 a 2 I the case a, we get the iverse Fioacci matrix, which is the kow result from 7] Corollary 25 The iverse Fioacci matrix F f i, ],, ) is equal to, i 2, f i,, i,, otherwise I the followig theorem we study rak of the matrix U a,,s) ): Theorem 22 Matrices U a,,s) of the order > 2 of a aritrary type s > or s < are sigular The geeralized ad U a,,) are regular Fioacci matrices U a,,s) 2 are always regular I the case matrices U a,,) Proof I the case s < the proof is trivial, sice s diagoal parallels elow the mai diagoal i U a,,s) are filled ) s < Deote y R i the i-th I the case s the last s rows ie the rows R s,, R ) i U a,,s) are completely For these rows it is ot difficult to verify from 23) y zeros ie the last s colums are zero colums), ad therefore raku a,,s) row of the matrix U a,,s) filled y the elemets U a,) i R i AR i B R i2, i s 2,, Therefore, etwee the rows R s,, R there is oly oe liearly idepedet row i the case s, ad oly two liearly idepedet i the case s > O the other had, it is clear that rows R,, R s are liearly idepedet Hece, i the case s > raku a,,s) ) { s <, s, 2, s > so the matrix U a,,s) is sigular From the previous argumetatio, it is ot difficult to verify that oth U a,,) matrices ad U a,,) are regular

8 P Staimirović et al / Discrete Applied Mathematics 56 28) Example 24 The last two colums of the matrix U a,,2) 4 A 2 a AB B A 3 a A 2 B 2A B ab 2 A 2 a AB B are the zero colums, ad raku a,,2) 4 ) 2 O the other had, the rak of the matrix U a,,2) 4 is 3, ecause of R 4 AR 3 B R 2 a a A B A ab a a A B A 2 ab A B A ab a A 3 ab A 2 2B A ab 2 A 2 ab A B A ab 3 Geeralized Fioacci matrix ad Pascal matrices Various types of Pascal matrices are ivestigated i,2,4,5,4,5] The geeralized Pascal matrix of the first kid P x] p x; )],,, is defied i 4]: ) i x i, i, p x; ) 3), i < I the case x, the geeralized Pascal matrix of the first kid reduces to ther well-kow Pascal matrix P p )],,,, which is defied i 3,4]: ) i, i, p ) 32), i < I the followig theorem we defie the matrix G x; a, ] g x; a, )],,, which gives a correlatio etwee the matrix U a,,) ad the geeralized Pascal matrix of the first kid: Theorem 3 The matrix G x; a, ] x, ), whose etries are defied y ) ) g x; a, ) x i ab A i 2 xi 2 x i satisfies i2 ) ik a2 B aa 2 ik a ik2 B ik x k k ) ] k, 33) P x] U a,,) G x; a, ] 34) Proof It is sufficiet to verify U a,,) P x] G x; a, ] It is evidet that g x; a, ) for i <, which is of the form 33) So, it remais to verify all the other cases The cases i ad i ca e simply verified:

9 264 P Staimirović et al / Discrete Applied Mathematics 56 28) g, x; a, ) u a,,), p x;, ) x ) x ; g, x; a, ) u a,,), p x;, ) u a,,), p x;, ) ab A 2 x ) )] 2 x x ab A 2 x I the last case, i 2, y applyig results of Theorem 2 we get g x; a, ) u a,,) i,i p x; ) u a,,) i,i p x; i, ) ) i xi i2 k which is also of the form 33) ab A 2 x i ) ik a2 B aa 2 ik i 2 ) i2 k a ik2 B ik x k u a,,) i,k p x; k, ) ) k, I the case A B we get aalogous result for the geeralized Fioacci matrix Corollary 3 The matrix G x; a, ] x, ), whose etries are defied y ) ) g x; a, ) x i i 2 i2 xi satisfies P x] F a,,) G x; a, ] a 2 x i k ) ik a2 a 2 ik a ik2 x k Moreover, the last corollary produces a kow result from 8] i partial case a, ad x : Corollary 32 Let M e the matrix with elemets defied y ) ) ) i i 2 i 3 m i The Pascal matrix ad the Fioacci matrix are related with P F M Proof The proof follows from M G ;, ] I the case a 2,, from Corollary 3 we give a correspodig result for Lucas matrices 6]: ) ] k, Corollary 33 The geeralized Pascal matrix of the first kid ad the Lucas matrix satisfy P x] L G x; 2, ], where ) ) i i 2 g x; 2, ) x x i 3x i 5) i i2 ) ] i2 k x ) k 2 ) k 2 After the sustitutio x i the previous result, the followig result immediately follows: Corollary 34 The Pascal matrix ad the Lucas matrix satisfy P L G ; 2, ], where ) ) i i 2 i2 ) i2 k g ; 2, ) 3 52) 2) k k k

10 P Staimirović et al / Discrete Applied Mathematics 56 28) I the followig theorem we defie the matrix H x; a, ] h x; a, )],,, which gives a similar correlatio etwee the matrix U a,,) ad the geeralized Pascal matrix of the first kid: Theorem 32 The matrix H x; a, ], ), defied y ) h x; a, ) x i i x ab A 2 x satisfies ) k a2 B aa 2 a k 2 B k x k k ) ] i 35) k P x] H x; a, ]U a,,) 36) Proof Similar as the proof of Theorem 3 A aalogous result for the geeralized Fioacci matrix ca e derived i the case A B Corollary 35 The matrix H x; a, ], ), defied y ) h x; a, ) x i i x a ) i 2 x ) k a2 a 2 ) ] i a k 2 x k k k satisfies P x] H x; a, ]F a,,) A aalogous result for Lucas matrices is 6]: Corollary 36 The Lucas matrix satisfies P x] H x; 2, ]L, where ) ) i i h x; 2, ) x x i 5x 3 ) 2 2 ) ] i ) k 2 k x k k The geeralized Pascal matrix of the secod kid Q x] q x; )],,, is defied y 4]: ) i x i 2, i, q x; ), i < 37) Theorem 33 The matrices S x; a, ] s x; a, )] ad T x; a, ] t x; a, )],,,, ) whose etries are defied y ) ) s x; a, ) x i ab A i 2 xi2 2 x i3 i2 ) ik a2 B aa 2 ik k a ik2 B ik x k2 ) ] k, 38)

11 266 P Staimirović et al / Discrete Applied Mathematics 56 28) ) ) t x; a, ) x i i x 2 ab A i 2 x satisfy Q x] U a,,) S x; a, ], Q x] T x; a, ]U a,,) ) k a2 B aa 2 a k 2 B k x k2 k ) ] i 39) k 3) 3) Proof Similar as the proof of Theorem 3 Corollary 37 The matrices S x; a, ] s x; a, )] ad T x; a, ] t x; a, )],,,, ) whose etries are defied y ) s x; a, ) x i xi2 a ) i 2 2 x i3 satisfy i2 ) ik a2 a 2 ik k t x; a, ) x i x 2 Q x] F a,,) S x; a, ], Q x] T x; a, ]F a,,) ) i a ik2 x k2 a 2 x ) k a2 a 2 a k ) i k 2 x k2 ) ] k, ) ] i k Theorem 34 I the case the matrix G a ; a, ] is defied y g a ; a, ) a)i 2 i ad satisfies P a ] F a,,) G a ] ; a, ) i a 2 a )a ) i 2 a 2 a 2 ) )] i 2 32) 33) Proof Follows from Corollary 3 ad the followig simple comiatorial idetity: i2 ) ) k i 2 k I a similar way as Theorem 34, the followig result ca e proved: Theorem 35 The matrix S a ; a, ] ) is defied y s a ; a, ) a)i 4 i ) i a 2 a 2 a) i ) )] i )

12 P Staimirović et al / Discrete Applied Mathematics 56 28) ad satisfies Q a ] F a,,) S a ] ; a, 35) I the partial case a 2, Theorems 34, 35 ad Corollary 37 yield the followig results: Corollary 38 The Lucas matrix satisfies: P 2] L G 2; 2, ], Q 2] L S 2; 2, ], P 2] H 2; 2, ]L, Q 2] T 2; 2, ]L, where ) ) )] i i 2 i 2 g 2; 2, ) 2) 4 i 2 6 5, ) ) )] i i 2 i 2 s 2; 2, ) 2) 4 i 4 6, ) ) i i ) h 2; 2, ) 2) 4 ] i i 2 6 5, k t 2; 2, ) 2) i 4 4 ) i 6 i ) ) i 5 5 k 3 ) ] i 2 2 2k k 4 Some comiatorial idetities I this sectio we ivestigate some comiatorial idetities ivolvig the geeralized Fioacci umers Theorem 4 If are positive itegers satisfyig i 2, ad, we have a ) i i ) Fa,) i a 2 F a,) )a i 2 ) k a )a 2 Proof From 32) we derive the followig idetities: g, a ; a, ), g, a ; a, ) a) 2 ) a 2 a )a ) k 2 F a,) ik a 2 a Now, the proof ca e derived y applyig idetities 42) ad the ext idetity p a ) ; a ) ) i i, i,, i < together with 32), 33) ad 2) a) k 2 k ) a2 a 2 k 2 2 )] 4) )a 2 42)

13 268 P Staimirović et al / Discrete Applied Mathematics 56 28) Theorem 42 If are positive itegers satisfyig i 2 ad, we have a ) ) i 2 i F a,) a 2 2 a 2 2 i Fa,) 2 i 2 )a ] ) ) k k k 2 a 2 a 2 a) 2 Proof From 34) we derive the followig idetities: s, a ) ; a, a2 2 2, F a,) ik a) k 4 k )] 43) s, a ) ; a, a2 2 2 )a ] 44) Now, the proof ca e derived y usig 44), the ext idetity q a ; ) ad 34), 35) ad 2) a ) ) i 2 i, i, i < Theorem 43 For r ad we have ) F a,) l r l r lr ) a 2 ) l 2 r k l2 kr ) lk a2 a 2 lk Proof I the partial case x from Corollary 3 we get g ; a, ) ) i a ) i 2 i2 2 ) ik a2 a 2 ik Now, the proof follows from ) p, r) r lr F a,) l g l,r ; a, ) I the partial case a, Theorem 43 reduces to Corollary 22 from 8] Corollary 4 For r ) k 3)! rk ) 2r ) k r) 2) F k r r )!k r)! kr a ik2 a lk2 ) k ) ] k r 45) Proof The proof ca e completed usig Theorem 43 ad Corollary 32, i the same way as i 8] 5 Coclusio I the preset paper we itroduce the matrix U a,,s) of type s, whose etries are umers U a,) satisfyig the geeral secod order o-degeerated recurrece formula U AU BU, δ A 2 4B real, ad iitial coditios U a,) a, U a,) I the case A B we defie the geeralized Fioacci matrix F a,,s) of type s, whose etries are geeralized Fioacci umers satisfyig kow recursive formula ad iitial coditios F a,) a, F a,) We oserve two regular cases s ad s ) of these matrices Geeralized Fioacci

14 P Staimirović et al / Discrete Applied Mathematics 56 28) matrices of type s or s correspod to kow defiitio of the usual Fioacci matrix, i the case a, I the case a 2,, s we otai defiitio of the Lucas matrix from 6] Iversio of the matrix U a,,s) ad geeralized Fioacci matrix is cosidered I certai cases we get kow results from 7,,6] A correlatio etwee the geeralized Fioacci matrix ad the Pascal matrix of the first ad the secod kid is cosidered I two partial cases a,, s ad a,, s ) we get kow result from 8] We get some comiatorial idetities ivolvig geeralized Fioacci umers I the partial case a 2,, s we derive aalogous result for Lucas matrices, itroduced i 6] Refereces ] R Aggarwala, MP Lamoureux, Ivertig the Pascal matrix plus oe, Amer Math Mothly 9 22) ] A Ashrafi, PM Giso, A ivolutory Pascal matrix, Liear Algera Appl ) ] R Brawer, M Pirovio, The liear algera of Pascal matrix, Liear Algera Appl ) ] GS Call, DJ Vellma, Pascal matrices, Amer Math Mothly 993) ] Gi-Cheo, Ji-Soo Kim, Stirlig matrix via Pascal matrix, Liear Algera Appl 329 2) ] AF Horadam, A geeralized Fioacci sequece, Amer Math Mothly 68 96) ] Gwag-Yeo Lee, Ji-Soo Kim, Sag-Gu Lee, Factorizatios ad eigevalues of Fioaci ad symmetric Fioaci matrices, Fioacci Quart 4 22) ] Gwag-Yeo Lee, Ji-Soo Kim, Seog-Hoo Cho, Some comiatorial idetities via Fioacci umers, Discrete Appl Math 3 23) ] D Kalma, R Mea, The Fioacci umers exposed, Math Magazie 76 23) 67 8 ] T Koshy, Fioacci ad Lucas Numers with Applicatios, Wiley, New York, 2 ] P Stǎicǎ, Cholesky factorizatios of matrices associated with r-order recurret sequeces, Electro J Comi Numer Theory 5 2) 25) #A6 2] JE Walto, AF Horadam, Some further idetities for the geeralized Fioacci sequece {H }, Fioacci Quart 2 974) ] Eric W Weisstei, Lucas Numer, i: MathWorld A Wolfram We Resource 4] Z Zhag, The liear algera of geeralized Pascal matrix, Liear Algera Appl ) 5 6 5] Z Zhag, J Wag, Beroulli matrix ad its algeraic properties, Discrete Appl Math 54 26) ] Z Zhag, Y Zhag, The Lucas matrix ad some comiatorial idetities, Idia J Pure Appl Math i press)