D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 9 3 IS S N 1 3 0 3 –5 9 9 1
ON THE GENERALIZED PERRIN AND CORDONNIER MATRICES
ADEM ¸SAH·IN
Abstract. In the present paper, we study the associated polynomials of Per-rin and Cordonnier numbers. We de…ne generalized PerPer-rin and Cordonnier matrices using these polynomials. We obtain the inverse of generalized Cor-donnier matrices and give some relationships between generalized Perrin and Cordonnier matrices. In addition, we give a factorization of generalized Cor-donnier matrices. Finally, we give some determinantal representation of asso-ciated polynomials Cordonnier numbers.
1. Introduction
There are several hundreds of papers on Fibonacci numbers and other recur-rence related sequences published during the last 30 years. Perrin numbers and Cordonnier numbers are some of them. Perrin numbers and Cordonnier numbers are
Pn = Pn 2+ Pn 3 for n > 3 and P1= 0; P2= 2; P3= 3;
Cn= Cn 2+ Cn 3 for n > 3 and C1= 1; C2= 1; C3= 1;
respectively.
The characteristic equation associated with the Perrin and Cordonnier sequence is x3 x 1 = 0 with roots ; ; ; in which = 1; 324718, is called plastic
number and lim n!1 Cn+1 Cn = lim n!1 Pn+1 Pn = :
The plastic number is used in art and architecture. Richard Padovan studied on plastic number in Architecture and Mathematics in [20, 21]. Christopher Bartlett found a signi…cant number of paintings with canvas sizes that have the aspect ratio of approximately 1.35. This ratio reminds Plastic number[1]. In [17] authors constructed the Plastic number in a heuristic way, explaining its relation to human
Received by the editors: May 15, 2016, Accepted: Sept. 25, 2016.
2010 Mathematics Subject Classi…cation. Primary 5A24, 11B68; Secondary 15A09, 15A15. Key words and phrases. Associated polynomials of Perrin and Cordonnier numbers, generalized Perrin and Cordonnier matrices.
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis t ic s .
perception in three-dimensional space through architectural style of Dom Hans van der Laan.
In [22], authors de…ned associated polynomials of Perrin and Cordonnier se-quences as;
Pn(x) = x2Pn 2(x) + Pn 3(x) for n > 3 and P1(x) = 0; P2(x) = 2; P3(x) = 3x;
Cn(x) = x2Cn 2(x) + Cn 3(x) for n > 3 and C1(x) = 1; C2(x) = x; C3(x) = x2;
respectively.
If we take x = 1 we obtain Pn(x) = Pn and Cn(x) = Cn.
Yilmaz and Taskara [26] developed the matrix sequences that represent Padovan and Perrin numbers. Kaygisiz and Bozkurt [5] de…ned k sequences of generalized order-k Perrrin numbers. Kaygisiz and Sahin [9] de…ned generalized Van der Laan and Perrin Polynomials, and generalizations of Van der Laan and Perrin Numbers. Many researchers have studied Linear Algebra of the some matrices. In [2] au-thors discussed the Linear Algebra of the Pascal Matrix, in [14] auau-thors examined the linear algebra of the k-Fibonacci matrix and the symmetric k-Fibonacci matrix. In [12] authors studied on the Pell Matrix. Sahin [23] gave the (q; x; s)-Fibonacci and Lucas matrices, obtained the inverse of these matrices and give some factor-ization of these matrices. Lee et al. [13] de…ned Fibonacci matrices and gave the factorization of Fibonacci matrix and obtained inverse of this matrix.
In addition many researchers have studied matrix representations of number sequences. Yilmaz and Bozkurt [25] gave matrix representation of Perrin sequences. Kaygisiz and Sahin [10] calculated terms of associated polynomials of Perrin and Cordonnier numbers by using determinants and permanents of various Hessenberg matrices. More examples can be found in [3, 6, 7, 8, 11, 15, 16, 19, 23].
Lemma 1.1. (Cf. Theorem of [3]) Let An be an n n lower Hessenberg matrix
for all n 1 and de…ne det(A0) = 1. Then, det(A1) = a11 and for n 2
det(An) = an;ndet(An 1) + n 1 X r=1 [( 1)n ran;r( n 1Y j=r aj;j+1) det(Ar 1)]:
In this paper, …rst we de…ne generalized Perrin and Cordonnier matrices using associated polynomials of Perrin and Cordonnier numbers. We obtain the inverse of generalized Cordonnier matrices with the aid of determinants of some Hessen-berg matrices which obtained from a part of these matrices. We also give some relationships between generalized Perrin and Cordonnier matrices in this section. Secondly, we give a factorization of generalized Cordonnier matrices. In the last section we give some determinantal representation of associated polynomials Cor-donnier numbers.
2. Generalized Perrin and Cordonnier matrices
De…nition 2.1. Let n be any positive integer, the n n lower triangular generalized Cordonnier matrix Cn;x= [ci;j]i;j=1;2;:::;n are de…ned by
ci;j= Ci j+1(x); if i j> 0 0; otherwise. (1) For example, C4;x= 2 6 6 4 1 0 0 0 x 1 0 0 x2 x 1 0 1 + x3 x2 x 1 3 7 7 5 :
De…nition 2.2. Let n be any positive integer, the n n lower triangular generalized Perrin matrix Pn;x= [pi;j]i;j=1;2;:::;n are de…ned by
pi;j= Pi j+2(x); if i j> 0 0; otherwise. (2) For example, P4;x= 2 6 6 4 2 0 0 0 3x 2 0 0 2x2 3x 2 0 2 + 3x3 2x2 3x 2 3 7 7 5 :
De…nition 2.3. Let n be any positive integer, the n n lower Hessenberg matrix sequenceHCn;x= [ai;j]i;j=1;2;:::;nare de…ned by
ai;j= Ci j+2(x); if i j + 1 0
0; otherwise. (3)
Lemma 2.4. Let c0(x) = 1, c1(x) = x; cn+1(x) = xcn(x)+
Pn
k=1( 1)n k+1Cn k+3(x)ck 1(x):
Then, det(HCn;x) = cn(x) for any positive integer n 1:
Proof. We proceed by induction on n. The result clearly holds for n = 1. Now suppose that the result is true for all positive integers less than or equal to n. We prove it for n + 1. In fact, by using Lemma 1.1 we have
det(HCn+1;x) = x det(HCn;x) + n X i=1 2 4( 1)n+1 ia n+1;i n Y j=i aj;j+1det(HCi 1;x) 3 5 = x det(HCn;x) + n X i=1 ( 1)n+1 iCn i+3(x) det(HCi 1;x) :
From the hypothesis of induction, we obtain
det(HCn+1;x) = xcn(x) +
n
X
i=1
Therefore, det(HCn;x) = cn(x) holds for all positive integers n.
Example 2.5. We obtain c3(x); c4(x) by using Lemma 2.4.
det 2 4 xx2 x1 01 1 + x3 x2 x 3 5 = 1 = c3(x);det 2 6 6 4 x 1 0 0 x2 x 1 0 1 + x3 x2 x 1 x + x4 1 + x3 x2 x 3 7 7 5 = x = c4(x):
Corollary 2.6. Let (HCn;x) be the n n Hessenberg matrix in (3). Then, det(HCn;x) =
cn(x) = xn 3 for any positive integer n 3:
Proof. We proceed by induction on n. The result clearly holds for n = 3. Now suppose that the result is true for all positive integers less than or equal to n. We prove it for n + 1. It is clear by the Laplace expansion of the last column that,
det(HCn+1;x) = x det(Mn;n) det(Mn 1;n)
= x det(HCn;x) det(Mn 1;n)
= xxn 3 det(Mn 1;n)
and since nth row of Mn 1;nis equal x2((n 1)th row of Mn 1;n) + ((n 2)th row
of Mn 1;n); det(Mn 1;n) = 0, where Mi;j is the (i; j) minor matrix ofHCn;x. So
we obtain
det(HCn+1;x) = xn 2:
Theorem 2.7. Let n be any positive integer, Cn;x is n n lower triangular
gener-alized Cordonnier matrix in (1) and (HCn;x) be the n n Hessenberg matrix in
(3). Then (Cn;x) 1= [cpi;j] is obtained by
[cpi;j] = 8 < : ( 1)i jdet( HCi j;x); if i j > 0 1; if i j = 0 0; otherwise.
Proof. Note that it su¢ ces to prove that Cn;x(Cn;x) 1= In. We take Cn;x(Cn;x) 1=
[ai;j]1 i;j n. It is obvious that ai;j =
Pn
k=0ci;kcpk;j = 0 for i j < 0 and ai;j =
Pn
k=0ci;kcpk;j= ci;icpi;i= 1 for i = j: For i > j 1 we have
ai;j = n X k=0 ci;kcpk;j = i X k=j ci;kcpk;j = Ci j+1(x) Ci j(x)c1(x) + + C1(x)( 1)i jci j(x)
and we know Ci j+1(x) =Pi js=1( 1)s+1cs(x)Ci j+1 s(x) from de…nition of cn(x).
Thus, we obtain ai;j =
Pn
k=0ci;kcpk;j = 0 for i > j 1 which implies that
Now, we show a relation between the generalized Perrin and Cordonnier matrices. The n n lower triangular matrix Tn;x:= [ri;j], (1 i; j n) is de…ned by
ri;j=
(Pi j
k=0( 1)kPi j k+2(x)ck(x); if i> j;
0; otherwise:
Theorem 2.8. Let n be any positive integer, Pn;x and Cn;xare n n lower
trian-gular generalized Perrin and Cordonnier matrices, then Tn;xCn;x= Pn;x:
Proof. Note that it su¢ ces to prove that Pn;x(Cn;x) 1 = Tn;x. It is obvious that
ri;j= 0 for i j > 0: For i j 1 we have n X k=0 pi;kcpk;j = n X k=0 Pi k+2(x)( 1)k jck j(x) = i X k=j Pi k+2(x)( 1)k jck j(x) = i j X k=0 Pi+j k+2(x)( 1)kck(x) = ri;j
which implies that Pn;x(Cn;x) 1= Tn;x; as desired.
Example 2.9. We obtain relation between the generalized Perrin and Cordonnier matrices for n=5 by using Theorem 2.8.
T5;xC5;x = P5;x: 2 6 6 6 6 4 2 0 0 0 0 x 2 0 0 0 x2 x 2 0 0 x3 x2 x 2 0 x4 x3 x2 x 2 3 7 7 7 7 5 2 6 6 6 6 4 1 0 0 0 0 x 1 0 0 0 x2 x 1 0 0 1 + x3 x2 x 1 0 x + x4 1 + x3 x2 x 1 3 7 7 7 7 5 = 2 6 6 6 6 4 2 0 0 0 0 3x 2 0 0 0 2x2 3x 2 0 0 3x3+ 2 2x2 3x 2 0 3x + 2x4 3x3+ 2 2x2 3x 2 3 7 7 7 7 5: Now we give a companion matrix Qn;x as follows:
Qn;x= 2 6 6 6 6 6 6 6 6 4 0 1 0 0 0 0 0 . .. 0 0 0 0 0 1 0 0 0 0 . .. 0 1 0 0 0 0 0 1 ( 1)n+1cn(x) ( 1)ncn 1(x) c3(x) 0 x 3 7 7 7 7 7 7 7 7 5 n n :
Theorem 2.10. Let m n be the integers, then the last column of matrix (Qn;x)m is 2 6 6 6 4 Cm n+2(x) .. . Cm(x) Cm+1(x) 3 7 7 7 5:
Proof. We proceed by induction on m. The result clearly holds for m = 1. Now suppose that the result is true for all positive integers less than or equal to m. We prove it for m + 1. The last column of matrix (Qn;x)m+1is
2 6 6 6 6 6 6 6 6 4 0 1 0 0 0 0 0 . .. 0 0 0 0 0 1 0 0 0 0 . .. 0 1 0 0 0 0 0 1 ( 1)n+1c n(x) ( 1)ncn 1(x) c3(x) 0 x 3 7 7 7 7 7 7 7 7 5 2 6 6 6 4 Cm n+2(x) .. . Cm(x) Cm+1(x) 3 7 7 7 5 = 2 6 6 6 4 Cm n+3(x) .. . Cm+1(x) ( 1)n+1c n(x)Cm n+2(x) + + c1(x)Cm+1(x) 3 7 7 7 5= 2 6 6 6 4 Cm n+3(x) .. . Cm+1(x) Cm+2(x) 3 7 7 7 5:
2.1. Factorizations of Generalized Cordonnier matrices. The set of all n-square matrices is denoted by Hn: A matrix H 2 Hn of the form
H = 2 6 6 6 6 4 H11 0 0 0 H22 . .. ... .. . . .. ... 0 0 0 Hkk 3 7 7 7 7 5 in which Hii 2 Hni (i = 1; 2; :::; k) and k P i=1
ni = n, is called block diagonal.
No-tationally, such a matrix is often indicated as H = H11 H22 Hkk; this is
called the direct sum of the matrices H11; H22; ; Hkk:
Lee et al. [13, 14] and Sahin[23] gave some factorization. Like these we consider factorization of Cn;x. Let In be the identity matrix of order n. We de…ne the
matrices Cn;x= [1] (Cn 1;x) and Dn= 2 6 6 6 4 1 0 0 c1(x) .. . In+2 ( 1)n+1c n+2(x) 3 7 7 7 5: (4) Lemma 2.11. (Ck;x)(Dk 3) = Ck;x for k 3:
Proof. We take (Ck;x) = [ci;j], (Dk 3) = [di;j] and Ck;x = [ci;j] and obtain k
P
s=1
ci;sds;j for i; j = 1; 2; :::; k: It is obvious from matrix product and de…nition
of In+2that c11= 1, k
P
s=1
ci;sds;j= ci;j for i = 1; 2; :::; k and j = 2; :::; k: For j = 1;
ci;1= k X s=1 ci;sds;1= Ci 1(x)c1(x) ( 1)k 1C1(x)ci 1(x): Using ck 1(x) = xck 2(x) C3(x)ck 3(x) + + ( 1)k 2Ck(x)c0(x) and c0(x) =
C1(x) = 1; we obtain Ck(x) = Ck 1(x)c1(x) + ( 1)k 1ck 1(x). So using these
last two equation the equation ci;1= Ci 1(x)c1(x) + ( 1)i 1C1(x)ci 1(x) =
Ci(x) is obtained.
Theorem 2.12. Let n 3 be any positive integer. Then Cn;x= (In 2 (D 1)) (I1
(Dn 4))(Dn 3).
Proof. From Lemma 2.11 and matrix product we obtain
Cn;x = (Cn;x)(Dn 3) = [(I1 Cn 1;x)(I1 (Dn 4))](Dn 3)
= [(I2 Cn 2;x)(I2 (Dn 5))](I1 (Dn 4))(Dn 3)
.. .
= (In 3 C3;x)(In 3 (D0)) (I1 (Dn 4))(Dn 3)
and (In 3 C3;x) = (In 2 (D 1)). Thus, we obtain
Cn;x= (In 2 (D 1))(In 3 (D0)) (I1 (Dn 4))(Dn 3):
Example 2.13. We give a factorization for C5;x by using Theorem 2.12:
(I3 D 1)(I2 D0)(I1 D1)D2 = 2 6 6 6 6 4 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 x 1 3 7 7 7 7 5 2 6 6 6 6 4 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 x 1 0 0 0 0 0 1 3 7 7 7 7 5 2 6 6 6 6 4 1 0 0 0 0 0 1 0 0 0 0 x 1 0 0 0 0 0 1 0 0 1 0 0 1 3 7 7 7 7 5 2 6 6 6 6 4 1 0 0 0 0 x 1 0 0 0 0 0 1 0 0 1 0 0 1 0 x 0 0 0 1 3 7 7 7 7 5
= 2 6 6 6 6 4 1 0 0 0 0 x 1 0 0 0 x2 x 1 0 0 x3+ 1 x2 x 1 0 x + x4 x3+ 1 x2 x 1 3 7 7 7 7 5:
Lemma 2.14. Dn are the (n + 3) (n + 3) Hessenberg matrices in (4). Then
(Dn) 1= 2 6 6 6 4 1 0 0 c1(x) .. . In+2 ( 1)n+2c n+2(x) 3 7 7 7 5: Proof. The proof is obvious that matrix product.
Corollary 2.15. Let n 3 be any positive integer. Then (Cn;x) 1= (Dn 3) 1(I1
(Dn 4)) 1 (In 2 (D 1)) 1:
Proof. Proof is obvious that previous lemma and equations (Ik (Dn k 3)) 1=
Ik (Dn k 3) 1:
3. Determinantal representation of associated polynomials Cordonnier numbers
Sahin and Ramirez gave a method for determinantal representation of Convolved Lucas polynomials in [24]. Using similar method, we give determinantal represen-tation of Cn(x).
Theorem 3.1. Let n 1 be an integer; Cn(x) be the nth associated polynomials
Cordonnier numbers and A(x)n = [ai;j]i;j=1;2;:::;n be an n n Hessenberg matrix
de…ned as ai;j= 8 > < > : 1; if i j = 1; ( 1)i jc i j+1(x); if i j 0; otherwise: (5) Then det( A(x)n ) = Cn+1(x):
Proof. We proceed by induction on m. The result clearly holds for n = 1; det( A(x)1 ) = x = C2(x). Now suppose that the result is true for all positive integers less than or
equal to n 1. We prove it for n.
In fact, using Cramers rule we have Cn(x) = det( A(x)n 1)Cn+1(x) det( A(x)n ) ) Cn+1(x) = det( A(x)n )Cn(x) det( A(x)n 1) From the hypothesis of induction we obtain
det( A(x)n ) = Cn+1(x):
Therefore, det( A(x)n ) = Cn+1(x) holds for all positive integers n.
Example 3.2. We obtain the polynomial C7(x) by using Theorem 3.1.
det 2 6 6 6 6 6 6 4 x 1 0 0 0 0 0 x 1 0 0 0 1 0 x 1 0 0 x 1 0 x 1 0 x2 x 1 0 x 1 x3 x2 x 1 0 x 3 7 7 7 7 7 7 5 = 2x3+ x6+ 1:
Corollary 3.3. Let m n be the integers, en is nth row of the identity matrix In.
Then
en( A(x)n )(Qn;x)meTn = Cm+2(x):
Proof. Proof is obvious from matrix product, Theorem 2.10 and equation
A(x)n [Cm n+2(x) Cm+1(x)]T = [0 0 0 Cn+2(x)]T:
Example 3.4. We obtain the polynomial C7(x) by using Corollary 3.3.
e6( A(x)6 )(Q6;x)5eT6 = 2x3+ x6+ 1 = C7(x)
Theorem 3.5. Let n 1 be an integer; Cn(x) be the nth associated polynomials
Cordonnier numbers and +B(x)n = [bs;t]s;t=1;2;:::;n be an n n Hessenberg matrix
de…ned as bs;t= 8 > < > : i; if s t = 1; (i)s tcs t+1; if s t 0; otherwise: Then det(+Bn(x)) = Cn+1(x):
Proof. If we multiply the kth column by ( 1)( i)k and the jth row by ( 1)ij of
the matrix A(x)n , where i =p 1, then the determinant is not altered. Therefore
Example 3.6. We obtain the polynomial C6(x) by using Theorem 3.5. det 2 6 6 6 6 4 x i 0 0 0 0 x i 0 0 1 0 x i 0 ix 1 0 x i x2 ix 1 0 x 3 7 7 7 7 5= 2x 2+ x5
The permanent of a n-square matrix is de…ned by perA =P 2SnQni=1ai (i);
where the summation extends over all permutations of the symmetric group Sn (cf. [18]). There is a relation between permanent and determinant of a
Hessen-berg matrix (cf. [4, 7]). Then it is clear the following corollary.
Corollary 3.7. Let n 1 be an integer; Cn(x) be the nth associated polynomials
Cordonnier numbers, +A(x)n = [us;t]s;t=1;2;:::;n and B(x)n = [vs;t]s;t=1;2;:::;n be the
n n Hessenberg matrices de…ned as
us;t= 8 > < > : 1; if s t = 1; ( 1)s tc s t+1; if s t 0; otherwise and vs;t= 8 > < > : i; if s t = 1; (i)s tc s t+1; if s t 0; otherwise :
Where i =p 1. Then per(+A(x)n ) = per( Bn(x)) = Cn+1(x):
Corollary 3.8. Let n 1 be an integer, A(x)n be the n n Hessenberg matrix in
(5), Cn(x) is the nth associated polynomials Cordonnier numbers and
\A(x) n = 2 6 6 6 4 1 0 0 .. . ( A(x)n 1) 0 1 3 7 7 7 5: Then, ( \A(x)n ) 1= Cn;x:
Proof. Proof is obvious from Theorem 2.7.
Acknowledgements. The author would like to express their pleasure to the anonymous reviewer for his/her careful reading and making some useful comments which improved the presentation of the paper.
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Current address : Faculty of Education, Gaziosmanpa¸sa University, 60250 Tokat, TURKEY E-mail address : adem.sahin@gop.edu.tr, hessenberg.sahin@gmail.com
