The Adjacency-Jacobsthal-Hurwitz Type Numbers

Filomat 32:16 (2018), 5623–5632 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1816623D University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat

Om¨ ürDevecia, Yes¸im Ak üzüma

aDepartment of Mathematics, Faculty of Science and Letters, Kafkas University 36100, Turkey

Abstract. In this paper, we define the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind. Then we give the exponential, combinatorial, permanental and determinantal representations and the Binet formulas of the adjacency-Jacobsthal-Hurwitz numbers of the first and second kind by the aid of the generating functions and the generating matrices of the sequences defined.

1. Introduction

It is well-known that Jacobsthal sequence Jn is deﬁned recursively by the equation { }

Jn+1 = Jn + 2Jn 1 − for n > 0, where J0 = 0, J1 = 1. In [5], Deveci and Artun deﬁned the adjacency-Jacobsthal sequence as follows:

Jm n (mn + k) = Jm n (mn n + k + 1) + 2Jm n (k) , , − , for k 1, m 2 and n 4 with initial constants Jm n (1) = = Jm n (mn 1) = 0 and Jm n (mn) = 1. ≥ ≥ ≥ , ··· , − , It is easy to see that the characteristic polynomial of the adjacency-Jacobsthal sequence is

mn mn n+1 f (x) = x x − 2. − − Suppose that the (n + k)th term of a sequence is deﬁned recursively by a linear combination of the preceding k terms:

an+k = c0an + c1an+1 + + ck 1an+k 1 ··· − − where c0, c1,..., ck 1 are real constants. In [10], Kalman derived a number of closed-form formulas for the generalized sequence− by the companion matrix method as follows: Let the matrix A be deﬁned by

2010 Mathematics Subject Classification. Primary 11K31; Secondary 11C20, 15A15 Keywords. adjacency-Jacobsthal-Hurwitz sequence, Hurwitz matrix, representation Received: 12 June 2017; Revised: 02 December 2017; Accepted: 04 December 2017 Communicated by Ljubisaˇ D.R. Kocinacˇ This Project was supported by the Commission for the Scientific Research Projects of Kafkas University, the Project No. 2016-FM-23 Email addresses: [email protected] (Om¨ ur¨ Deveci), [email protected] (Yes¸im Akuz¨ um)¨ O.¨ Deveci, Y. Aküzüm / Filomat 32:16 (2018), 5623–5632 5624

 0 1 0 0 0     0 0 1 ··· 0 0     ···   ..  h i  0 0 0 . 0 0  A = ai,j =   k k  . . . . .  ×  . . . . .   . . . . .     0 0 0 0 1   ···  c0 c1 c2 ck 2 ck 1 − − , then

     a0   an       a1   an+1  An   =    .   .   .   .      ak 1 an+k 1 − − for n > 0. Let an nth degree real polynomial q be given by

n n 1 q (x) = c0x + c1x − + + cn 1x + cn. ··· − h i In [9], the Hurwitz matrix Hn = hi,j associated to the polynomial q was deﬁned as shown: n n ×

 c c c 0 0 0   1 3 5 ·········   . . .   . . .   c0 c2 c4 . . .     . . .   0 c1 c3 . . .     . . . .   . c c .. 0 . .   0 2   . . . .  Hn =  . . . .   . 0 c1 . cn . .     . . .. .   . . c0 . cn 1 0 .   −   . . .   . . .   . . 0 cn 2 cn .   −   . . .   . . . cn 3 cn 1 0   − −  0 0 0 cn 4 cn 2 cn ········· − − . Recently, many authors have studied number theoretic properties such as these obtained from homoge- neous linear recurrence relations relevant to this paper [3, 4, 6–8, 11, 12, 14, 16–19]. In this paper, we define the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind by using Hurwitz matrix for characteristic polynomial of the adjacency-Jacobsthal sequence of order 4m. Then we develop some their properties such as the generating function, exponential representations, the generating matrices and the combinatorial representations. Also, we give relationships among the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind and the permanents and the determinants of certain matrices which are produced by using the generating matrices of the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind. Finally, we obtain the Binet formulas for the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind by the aid of the roots of characteristic polynomials of the adjacency-Jacobsthal- Hurwitz sequences of the first and second kind. O.¨ Deveci, Y. Aküzüm / Filomat 32:16 (2018), 5623–5632 5625

2. The Main Results

It is readily seen that Hurwitz matrix for characteristic polynomial of the adjacency-Jacobsthal sequence J h i of order 4m, H = hi,j is deﬁned by 4m 4m 4m ×   1 if i = 2ı and j = ı for 1 ı 2m,   1 if i = 1 + 2ı and j = 2 + ı for≤ 1 ≤ı 2m 1, h =  i,j  −2 if i = 2ı and j = 2m + ı for 1≤ ı≤ 2m−,   −0 otherwise. ≤ ≤

J By the aid of the matrix H4m, we deﬁne the adjacency-Jacobsthal-Hurwitz sequences of the ﬁrst and second kind, respectively by:

J1 (4m + k) = J1 (2m + k) 2J1 (k) for k 1 and m 4, (1) m m − m ≥ ≥ where

J1 (1) = 1, J1 (2) = = J1 (2m) = 0, J1 (2m + 1) = 1, J1 (2m + 2) = = J1 (4m) = 0 m m ··· m m m ··· m and

J2 (4m + k) = J2 (k) 2J2 (2m + k) for k 1 and m 4, (2) m m − m ≥ ≥ where

J2 (1) = 1, J2 (2) = = J2 (4m 1) = 0, J2 (4m) = 1. m m ··· m − m Clearly, the generating functions of the adjacency-Jacobsthal-Hurwitz sequences of the ﬁrst and second kind are given by

1 11 (x) = 1 x2m + 2x4m − and

1 + 3x2m 12 (x) = , 1 + 2x2m x4m − respectively. It can be readily established that the adjacency-Jacobsthal-Hurwitz sequences of the ﬁrst and second kind have the following exponential representations, respectively:

 i  2m X∞ x i 1  2m  1 (x) = exp  1 2x   i −   i=1  and

 i  2m X∞ x i 2 2m  2m  1 (x) = 1 + 3x exp  x 2  .  i −   i=1  O.¨ Deveci, Y. Ak¨uz¨um / Filomat 32:16 (2018), 5623–5632 5626

By equations (1) and (2), we can write the following companion matrices, respectively:

(2m) th

 ↓   0 0 1 0 0 2   ··· ··· −   1 0 0 0 0 0 0   ···   0 1 0 0 0 0 0   ···   0 0 1 0 0 0 0   ···  C1  0 0 0 1 0 0 0  m =    . . . . . ··· .   ......   ......     ......   ......    0 0 0 0 0 1 0 4m 4m ··· × and    0 1 0 0 0 0 0   ···   0 0 1 0 0 0 0   ···   0 0 0 1 0 0 0   ···   . . . .  2  . . .. .  Cm =    ···   0 0 0 0 1 0 0   ···   0 0 0 0 0 1 0   ···   0 0 0 0 0 0 1   ···  1 0 ... 0 2 0 0 4m 4m. − ··· × (2m +↑ 1) th

1 2 The companion matrices Cm and Cm are called the adjacency-Jacobsthal-Hurwitz matrices of the ﬁrst and 1 second kind, respectively. For detailed information about the companion matrices, see [13, 15]. Let Jm (α) 2 1,α 2,α and Jm (α) be denoted by Jm and Jm . By mathematical induction on α, we derive

 1,α+1 1,α+2 1,α+2m 1,α 2m+1 1,α 1 1,α   Jm Jm Jm 2Jm − 2Jm − 2Jm     1,α 1,α+1 ··· 1,α+2m 1 − 1,α 2m · · · − 1,α 2 − 1,α 1   J J J − 2J − 2J − 2J −  α  m m m m m m  C1  ··· − · · · − −  ( m) = ......   ......   ......     1,α 4m+2 1,α 4m+3 1,α 2m+1 1,α 6m+2 1,α 4m 1,α 4m+1   J − J − J − J − J − J −  m m m 2 m 2 m 2 m 4m 4m ··· − · · · − − × (3) and

 2,α 2,α 1 2,α 2m+1 2,α+2m 2,α+2 2,α+1   Jm Jm − Jm − Jm Jm Jm   2,α+1 2,α ··· 2,α 2m+2 2,α+2m+1 ··· 2,α+3 2,α+2  α  J J J − J J J  2  m m ··· m m ··· m m  Cm =  ......  (4)  ......   ......   2,α+4m 1 2,α+4m 2 2,α+2m 2,α+6m 1 2,α+4m+1 2,α+4m  − − − Jm Jm Jm Jm Jm Jm 4m 4m. ··· ··· × α α for α 1. Note that det C1 = (2)α and det C2 = ( 1)α ≥ m m − Let K (k , k ,..., kv) be a v v companion matrix as follows: 1 2 ×    k1 k2 kv   ···   1 0 0  K (k , k ,..., k ) =   1 2 v  . .. .   . . .    0 1 0 ··· . O.¨ Deveci, Y. Ak¨uz¨um / Filomat 32:16 (2018), 5623–5632 5627

(n) n Theorem 2.1. (Chen and Louck [2]) The i, j entry ki,j (k1, k2,..., kv) in the matrix K (k1, k2,..., kv) is given by the following formula: ! X tj + tj+1 + + tv t + + t (n) ··· 1 v t1 tv ki,j (k1, k2,..., kv) = ··· k1 kv , (5) t1 + t2 + + tv × t1,..., tv ··· (t1,t2,...,tv) ···

t1+ +tv (t1+ +tv)! where the summation is over nonnegative integers satisfying t + 2t + + vtv = n i + j, ··· = ··· is a 1 2 t1,...,tv t1! tv! multinomial coefficient, and the coefficients in (5) are defined to be 1 if n···= i j. − ··· − Now we concentrate on finding combinatorial representations for the adjacency-Jacobsthal-Hurwitz numbers of the first and second kind.

Corollary 2.2. The following hold: 1 P tα+tα+1+ +t4m t1+ +t4m t4m (i) J (n) = ··· ··· ( 2) for 1 α 2m, m (t1,t2...,t4m) t +t + +t m t ,...,t m 1 2 ··· 4 × 1 4 − ≤ ≤ where the summation is over nonnegative integers satisfying t + 2t + + (4m) t m = n 1. 1 2 ··· 4 − 1 P t4m t1+ +t4m t4m (ii) J (n) = ··· ( 2) , m (t1,t2...,t4m) t +t + +t m t ,...,t m 1 2 ··· 4 × 1 4 − where the summation is over nonnegative integers satisfying t + 2t + + (4m) t m = n + 4m 1. 1 2 ··· 4 − 2 P tα+tα+1+ +t4m t1+ +t4m t2m+1 (iii) J (n) = ··· ··· ( 2) for 1 α 2m, m (t1,t2...,t4m) t +t + +t m t ,...,t m 1 2 ··· 4 × 1 4 − ≤ ≤ where the summation is over nonnegative integers satisfying t + 2t + + (4m) t m = n. 1 2 ··· 4 Proof. If we take i = α + 1, j = α such that 1 α 2m for case (i)., i = 1, j = 4m for case (ii) and i = j = α such that 1 α 2m for case (iii) in Theorem≤ 2.1 ,≤ then we can directly see the conclusions from equations (3) and (4). ≤ ≤

h i th Deﬁnition 2.3. A u v real matrix M = mi j is called a contractible matrix in the k column (resp. row.) if × , the kth column (resp. row) contains exactly two non-zero entries.

th Let u1, u2, ...,umn be row vectors of the matrix M. If M is contractible in the k column such that th mi,k , 0, mj,k , 0 and i , j, then the (u 1) (v 1) matrix Mij:k obtained from M by replacing the i row th − × −th th with mi,kxj +mj,kxi and deleting the j row. The k column is called the contraction in the k column relative to the ith row and the jth row. If M is a real matrix of order α > 1 and N is a contraction of M, then per (M) = per (N) which was proved in [1]. Now we consider relationships between the adjacency-Jacobsthal-Hurwitz sequences of the ﬁrst and second kind and the permanents of certain matrices which are obtained by using the generating matrices of these sequences. h i h i Let u > 4m be a positive integer and suppose that M1,u = m1,u,m and M2,u = m2,u,m are the u u m i,j m i,j × super-diagonal matrices, deﬁned by

(2m) th (4m) th ↓ ↓  0 0 1 0 0 2 0 0 0 0     1··· 0 0 1··· 0 −0 2··· 0 0 0     ··· ··· − ···   0 1 0 0 1 0 0 2 0 0   ··· ··· − ···   ......   ......  1,u   M =  0 0 1 0 0 1 0 0 2 0  m    0··· 0 0 1··· 0 0 1··· 0 −0 2     ··· ··· ··· −   0 0 0 0 1 0 0 1 0 0   ··· ··· ···   ......   ......    0 0 0 0 0 0 1 0 0 1 0 u u ··· ··· × O.¨ Deveci, Y. Ak¨uz¨um / Filomat 32:16 (2018), 5623–5632 5628 and  0 1 0 0 0 0 0 0 0 0 0 0   ··· ···   0 0 1 0 0 0 0 0 0 0 0 0     . . . ···. . ···. . .   ......   ......     0 0 0 1 0 0 0 0 0 0 0   ··· ··· ···   2 0 0 0 1 0 0 0 0 0 0   − ··· ··· ···   0 2 0 0 0 1 0 0 0 0 0     . −. . ···. . . ···. . ···. .  2,u  ......  M =  ......  m    0 0 2 0 0 0 1 0 0 0 0   ··· − ··· ···   1 0 0 2 0 0 0 1 0 0 0   ··· − ··· ···   0 1 0 0 2 0 0 0 1 0 0     . . . ···. −. . ···. . . ···. .   ......   ......     0 0 1 0 0 2 0 0 0 1 0   ··· ··· − ···   0 0 0 1 0 0 2 0 0 0 1   ··· ··· − ···  0 0 0 0 1 0 0 2 0 0 0 u u. ··· ··· − ··· × (u 4m↑+ 1) th (u 2m↑+ 1) th − − Theorem 2.4. For u > 4m, 1,u 1 2,u 2 per Mm = Jm (u + 1) and per Mm = Jm (u + 4m) .

1,u Proof. Let us consider the matrix Mm and the adjacency-Jacobsthal-Hurwitz sequence of the ﬁrst kind. We use induction on u. Now we assume that the equation holds for u 4m, then we show that the equation 1,u+1 ≥ holds for u + 1. If we expand the per Mm by the Laplace expansion of permanent according to the ﬁrst row, then we obtain 1,u+1 1,u 2m+1 1,u 4m+1 per M = per M − 2per M − . m m − m 1,u 2m+1 1 1,u 4m+1 1 Since per M − = J (u 2m + 2) and per M − = J (u 4m + 2) , it is easy to see that m m − m m − 1,u+1 1 1 1 per Mm = Jm (u 2m + 2) 2Jm (u 4m + 2) = Jm (u + 2). So we have the conclusion. − − − 2,u There is a similar proof for the matrix Mm and the adjacency-Jacobsthal-Hurwitz sequence of the second kind.

1,v h 1,v,mi 2,v h 2,v,mi Let v 4m be a positive integer and suppose that the matrices Am = a and Am = a are ≥ i,j v v i,j v v defined, respectively, by × ×   if i = ı and j = ı + 2m 1 for 1 ı v 2m + 1  − ≤ ≤ −  1 and  1,v,m  i = ı + 1 and j = ı for 1 ı v 2m, a =  ≤ ≤ − i,j  1 if i = 1 + ı and j = ı for v 2m + 1 ı v 1,  − − ≤ ≤ −  2 if i = ı and j = 4m + ı 1 for 1 ı v 4m + 1,   −0 otherwise− ≤ ≤ − and   if i = ı + 2m 1 and j = ı + 2m for 1 ı v 2m  − ≤ ≤ −  1 and  2,v,m  i = ı + 4m 1 and j = ı for 1 ı v 4m + 1, a =  − ≤ ≤ − i,j  1 if i = ı and j = ı + 1 for 1 ı 2m 1,  − ≤ ≤ −  2 if i = ı + 2m 1 and j = ı for 1 ı v 2m + 1,   −0− otherwise. ≤ ≤ − O.¨ Deveci, Y. Aküzüm / Filomat 32:16 (2018), 5623–5632 5629

Then we can give the permanental representations other than the above by the following theorem.

Theorem 2.5. For v 4m, ≥ per A1,v = J1 (v + 1) and per A2,v = J2 (v + 4m) . m − m m − m 2,v Proof. Let us consider the matrix Am and the adjacency-Jacobsthal-Hurwitz sequence of the second kind. The assertion may be proved by induction on v. Let the equation be hold for v 4m, then we show that the 2,v+1 ≥ equation holds for v + 1. If we expand the per Am by the Laplace expansion of permanent according to the ﬁrst row, then we obtain

2,v+1 2,v 4m+1 2,v 2m+1 per A = per A − 2per A − m m − m = J2 (v + 1) 2 J2 (v + 2m + 1) = J2 (v + 4m + 1) . − m − − m − m Thus we have the conclusion. 1,v There is a similar proof for the matrix Am and the adjacency-Jacobsthal-Hurwitz sequence of the ﬁrst kind.

Now we deﬁne a v v matrix Bv as in the following form: × m (v 4m) th −  ↓   1 1 0 0   − · · · − ···   1   − 1,v 1  v  0 A −  Bm =  m  ,  .   .   .   0  then we have the following result:

Corollary 2.6. For v > 4m + 1,

v 1 X− v 1 perBm = Jm (i) . i=1

v Proof. If we extend the perBm with respect to the ﬁrst row, we obtain

v v 1 1,v 1 per Bm = per Bm− + perAm − .

From Theorem 2.4, Theorem 2.5 and induction on v, the proof follows directly.

A matrix M is called convertible if there is an n n (1, 1)-matrix K such that det (M K) = perM, where M K denotes the Hadamard product of M and K.× − ◦ ◦ h i h i Now assume that the matrices T = ti,j and S = si,j are defined by u u v v × ×    1 1 1 1 1   ···   1 1 1 1 1   − ···   1 1 1 1 1   − ···  T =  ......   ......       1 1 1 1 1   1 ··· 1− 1 1 1  ··· − O.¨ Deveci, Y. Aküzüm / Filomat 32:16 (2018), 5623–5632 5630 and    1 1 1 1 1   − ···   1 1 1 1 1   − ···   . . . .   . . .. .  S =    1 1 1 1 1     ··· −   1 1 1 1 1   1 1 ··· 1 1− 1  ··· . Then we give relationships between the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind and the determinants of the Hadamard products M1,u T, A1,v T, M2,u S and A2,v S. m ◦ m ◦ m ◦ m ◦ Theorem 2.7. Let u, v 4m, then ≥ det M1,u T = J1 (u + 1) , m ◦ m det A1,v T = J1 (v + 1) , m ◦ − m det M2,u S = J2 (u + 4m) m ◦ m and det A2,v S = J2 (v + 4m) . m ◦ − m 1,u 1,u 1,v 1,v 2,u 2,u 2,v Proof. Since det Mm T = per Mm , det Am T = per Am , det Mm S = per Mm and det Am S ◦ ◦ ◦ ◦ = per A2,u for u, v 4m, by Theorem 2.4 and Theorem 2.5, we have the conclusion m ≥ Now we concentrate on finding the Binet formulas for the adjacency-Jacobsthal numbers. 1,u 2,u Clearly, the characteristic equations of the matrices Mm and Mm are x4m x2m + 2 = 0 − and x4m + 2x2m 1 = 0, − n (1) (1) (1) o respectively. It is easy to see that the above equations do not have multiple roots. Let β1 , β2 , . . . , β4m and n (2) (2) (2) o 1,u 2,u (λ) β1 , β2 , . . . , β4m be the sets of the eigenvalues of the matrices Mm and Mm , respectively and let Vm be (4m) (4m) Vandermonde matrix as follows: ×  4m 1 4m 1 4m 1   β(λ) − β(λ) − β(λ) −   1 2 4m   4m 2 4m 2 ··· 4m 2   (λ) − (λ) − (λ) −   β1 β2 β4m  (λ)  ···  V  . . .  m =  . . .  ,  . . .   (λ) (λ) (λ)   β β β   1 2 4m   1 1 1  ··· where λ = 1, 2. Now assume that

 (λ)α+4m i   β −   1   (λ)α+4m i   β −  (λ)  2  Wm (i) =  .   .   .   α+4m i   (λ) −  β4m (λ) (λ) (λ) (λ) and V i, j is a (4m) (4m) matrix obtained from V by replacing the jth column of V by W (i). m × m m m O.¨ Deveci, Y. Ak¨uz¨um / Filomat 32:16 (2018), 5623–5632 5631

Theorem 2.8. For α 1 and λ = 1, 2, ≥ (λ) det Vm i, j cm,λ,α = , i,j (λ) det Vm

α λ h m,λ,αi where Cm = ci,j .

4m 2m (1) (1) (1) Proof. Let us consider λ as 1. Since the equation x x +2 = 0 does not have multiple roots, β1 , β2 , . . . , β4m 1,u − 1 (1) (1) 1 are distinct and so the matrix Mm is diagonalizable. Then, it is readily seen that CmVm = Vm Ωm, where 1 1 (1) (1) (1) (1) (1) − 1 (1) 1 Ωm = β1 , β2 , . . . , β4m . Since the Vandermonde matrix Vm is invertible, we can write Vm CmVm = Ωm. α α Thus, we easily see that the matrix C1 is similar to Ω1 . Then, we have C1 V(1) = V(1) Ω1 for α 1. So m m m m m m ≥ we obtain the following linear system of equations:

 4m 1 4m 2 α+4m i  m,1,α (1) − m,1,α (1) − m,1,α (1) −  c β + c β + + c = β  i,1 1 i,2 1 ··· i,4m 1  m,1,α (1)4m 1 m,1,α (1)4m 2 m,1,α (1)α+4m i  c β − + c β − + + c = β −  i,1 2 i,2 2 ··· i,4m 2  .  .  .  4m 1 4m 2 α+4m i  cm,1,α β(1) − + cm,1,α β(1) − + + cm,1,α = β(1) − . i,1 4m i,2 4m ··· i,4m 4m m,1,α Then, for each i, j = 1, 2,..., 4m, we derive ci,j as

(1) det Vm i, j . (1) det Vm

There is a similar proof for λ = 2. As an immediate consequence of this we have

Corollary 2.9. For α 1, ≥ det V1 (k + 1, k) 1 m Jm (α) = for 1 k 2m , (1) ≤ ≤ det Vm det V1 (1, 4m) 1 m Jm (α) = , − (1) 2 det Vm and det V2 (k, k) 2 m Jm (α) = for 1 k 2m. (2) ≤ ≤ det Vm

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