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Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp 1065-1078.

DETERMINANTS AND PERMANENTS OF HESSENBERG MATRICES AND GENERALIZED LUCAS POLYNOMIALS ˙ K. KAYGISIZ∗ AND A. S ¸ AHIN

Communicated by Amir Daneshgar Abstract. In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p-polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the conditions under which the determinants of the Hessenberg matrix become its permanents.

1. Introduction Fibonacci numbers fn , Lucas numbers ln and Perrin numbers rn are defined by fn = fn−1 + fn−2 for n > 2 and f1 = f2 = 1, ln = ln−1 + ln−2 for n > 1 and l0 = 2, l1 = 1, rn = rn−2 + rn−3 for n > 3 and r0 = 3, r1 = 0, r2 = 2, respectively. MSC(2010): Primary: 11B37; Secondary: 15A15, 15A51. Keywords: Generalized Lucas polynomials, generalized Perrin polynomials, Hessenberg matrix, determinant and permanent. Received: 11 May 2012, Accepted: 12 August 2012. ∗Corresponding author c 2013 Iranian Mathematical Society. ⃝ 1065

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Kaygısız and S ¸ ahin

Generalizations of these sequences have been studied by a number of researchers. For instance; Miles [16] defined generalized order-k Fibonacci numbers, Er [2] defined k sequences of generalized order-k Fibonacci numbers and Kaygısız and Bozkurt [4] defined k-generalized order-k Perrin numbers. MacHenry [12] defined generalized Fibonacci polynomials Fk,n (t) and Lucas polynomials Gk,n (t) as follows; Fk,n (t) = 0, n < 0, Fk,0 (t) = 1, k ∑ Fk,n (t) = tj Fk,n−j (t), j=1

Gk,n (t) = 0, n < 0, Gk,0 (t) = k, Gk,1 (t) = t1 , Gk,n (t) = Gk−1,n (t), 1 ≤ n < k, k ∑ Gk,n (t) = tj Gk,n−j (t), n ≥ k j=1

where ti (1 ≤ i ≤ k) are constant coefficients of the core polynomial P (x; t1 , t2 , . . . , tk ) = xk − t1 xk−1 − · · · − tk . In [13, 14], authors obtained some properties of this polynomials. In addition, in [15], authors obtained (n, k ∈ N, n ≥ 1) (1.1)

∑ n ( |a| ) Gk,n (t) = ta11 . . . takk |a| a1,..., ak a⊢n

where ai are nonnegative integers for all i (1 ≤ i ≤ k), with initial conditions given by Gk,0 (t) = k, Gk,−1 (t) = 0, · · · , Gk,−k+1 (t) = 0. Throughout this paper, the notations a ⊢ n and |a| are used instead k k ∑ ∑ of jaj = n and aj , respectively. j=1

j=1

Kaygısız and S¸ahin [5] defined generalized Perrin polynomials Rk,n (t) by using generalized Lucas polynomials. The generalized bivariate Lucas p-polynomials [20] are defined by Lp,n (x, y) = xLp,n−1 (x, y) + yLp,n−p−1 (x, y) for n > p, with boundary conditions Lp,0 (x, y) = (p+1), Lp,n (x, y) = xn , n = 1, 2, ..., p.

Determinants and permanents of Hessenberg matrices

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Table 1. Cognate polynomial sequences 1. k t1 ti (2 ≤ i ≤ (k − 1)) tk k 0 ti tk k x 0 y 3 0 1 1

Gk,n (t) Rk,n (t) Lp,n (x, y) rn

Table 2. [20]Cognate polynomial sequences 2. x y p x y 1 x 1 p x 1 1 1 1 p 1 1 1 2x y p 2x y 1 2x 1 p 2x 1 1 2 1 1 2x −1 1 x 2y p x 2y 1 1 2y 1 1 2 1

Lp,n (x, y) bivariate Lucas polynomials Ln (x, y) Lucas p-polynomials Lp,n (x) Lucas polynomials ln (x) Lucas p-numbers Lp (n) Lucas numbers Ln bivariate Pell-Lucas p-polynomials Lp,n (2x, y) bivariate Pell-Lucas polynomials Ln (2x, y) Pell-Lucas p-polynomials Qp,n (x) Pell-Lucas polynomials Qn (x) Pell-Lucas numbers Qn Chebysev polynomials of the first kind Tn (x) bivariate Jacobsthal-Lucas p-polynomials Lp,n (x, 2y) bivariate Jacobsthal-Lucas polynomials Ln (x, 2y) Jacobsthal-Lucas polynomials jn (y) Jacobsthal-Lucas numbers jn

Tables 1 and 2 show that Gk,n (t) are general form of many sequences and polynomials. Therefore, any result obtained from the polynomials Gk,n (t) is valid for all sequences and polynomials mentioned in these tables. On the other hand, many researchers studied determinantal and permanental representations of k sequences of generalized order-k Fibonacci and Lucas numbers. For example, Minc [17] defined an n × n (0,1)matrix F (n, k) and showed that the permanents of F (n, k) is equal to the generalized order-k Fibonacci numbers. In [10, 11], authors defined two (0,1)-matrices and showed that the permanents of these matrices are the generalized Fibonacci and Lucas ¨ numbers. Ocal et al. [18] gave some determinantal and permanental representations of k-generalized Fibonacci and Lucas numbers and obtained Binet’s formulas for these sequences. Kılı¸c and Stakhov [8] gave

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permanent representation of Fibonacci and Lucas p-numbers. Kılı¸c and Ta¸scı [9] studied permanents and determinants of Hessenberg matrices. Yılmaz and Bozkurt [22] derived some relationships between Pell and Perrin sequences, as well as permanents and determinants of a type of Hessenberg matrices. Kaygısız and S¸ahin [7] gave some determinantal and permanental representations of Fibonacci type numbers. Kaygısız and S¸ahin [6] gave some determinantal and permanental representations of generalized bivariate Lucas p-polynomials. In [3, 19, 21], authors gave some relations between determinants and permanents. The main purpose of this paper is to give some determinantal and permanental representations of generalized Lucas polynomials by using various Hessenberg matrices. Then we provide some conditions under which the determinants of the Hessenberg matrix become its permanents.

2. The determinantal representations An n × n matrix + An = (aij ) is called lower Hessenberg matrix if aij = 0 when j − i > 1 i.e.,   a11 a12 0 ··· 0  a21  a22 a23 ··· 0    a31  a32 a33 ··· 0   (2.1) A =   .. .. .. .. + n   . . . .    an−1,1 an−1,2 an−1,3 · · · an−1,n  an,1 an,2 an,3 · · · an,n Lemma 2.1. [1] Let + An be the n × n lower Hessenberg matrix for all n ≥ 1 and define det(+ A0 ) = 1. Then, det(+ A1 ) = a11 and for n ≥ 2 (2.2) n−1 n−1 ∑ ∏ det(+ An ) = an,n det(+ An−1 ) + [(−1)n−r an,r ( aj,j+1 ) det(+ Ar−1 )]. r=1

j=r

Now, we define two Hessenberg matrices + Ck,m and determinants give the generalized Lucas polynomials.

− Ck,m

whose

Theorem 2.2. Let k ≥ 2 and n ≥ 1 be integers, and let Gk,n (t) be the generalized Lucas polynomials and + Ck,n = (crs ) an n × n Hessenberg


matrix, given by   i|r−s| . tr−s+1 (r−s) ,   t2 crs = i|r−s| . tr−s+1 (r−s) .(r − s + 1),  t2   0,

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if s ̸= 1 and − 1 ≤ r − s < k, if s = 1 and − 1 ≤ r − s < k, otherwise

i.e., 

+ Ck,n

t1 2i 3i2 tt23 2 .. .

       tk =  kik−1 k−1 t2   0    ..  . 0

where t0 = 1 and i =

it2 t1 i .. . tk−1 tk−2 2 tk ik−1 k−1 t2

0 0 it2 .. .

··· ··· ···

tk−3 tk−4 2 t ik−3 k−2 tk−3 2

···

0 it2 t1 .. . tk−2 tk−3 2 t ik−2 k−1 tk−2 2

ik−2

ik−3

.. . 0

.. . 0

ik−4

.. . ···

··· .. . i

0 0 0 .. . 0



           0    it2  t1

√ −1. Then,

(2.3)

det(+ Ck,n ) = Gk,n (t).

Proof. To prove (2.3), we use the mathematical induction on m. The result is true for m = 1 by hypothesis. Assume that it is true for all positive integers less than or equal to m, namely, det(+ Ck,m ) = Gk,m (t). Then, by using Lemma 2.1, we have det(+ Ck,m+1 ) = cm+1,m+1 [ det(+ Ck,m ) ] m m ∑ ∏ m+1−r + (−1) cm+1,r cj,j+1 det(+ Ck,r−1 ) r=1

= t1 det(+ Ck,m ) +

+

m ∑ r=m−k+2



m−k+1 ∑

j=r

 (−1)m+1−r cm+1,r

r=1

(−1)m+1−r cm+1,r

m ∏ j=r

m ∏ j=r

 cj,j+1 det(+ Ck,r−1 ) 

cj,j+1 det(+ Ck,r−1 )

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= t1 det(+ Ck,m )   m m ∑ ∏ (−1)m+1−r cm+1,r + cj,j+1 det(+ Ck,r−1 ) j=r

r=m−k+2

= t1 det(+ Ck,m )   m m ∑ ∏ t (−1)m+1−r .im+1−r m−r+2 + it2 det(+ Ck,r−1 ) (m−r+1) t j=r r=m−k+2 2 = t1 det(+ Ck,m ) ] [ m ∑ m+1−r tm−r+2 m+1−r (m−r+1) + .i .t2 det(+ Ck,r−1 ) (−i) (m−r+1) t r=m−k+2 2 = t1 det(+ Ck,m ) +

m ∑

tm−r+2 det(+ Ck,r−1 )

r=m−k+2

= t1 det(+ Ck,m ) + t2 det(+ Ck,m−1 ) + · · · + tk det(+ Ck,m−(k−1) ). From the hypothesis and definition of generalized Lucas polynomials we obtain det(+ Ck,m+1 ) = t1 Gk,m (t) + · · · + tk Gk,m−(k−1) (t) = Gk,m+1 (t). □

Therefore, (2.3) holds for all positive integers.

Example 2.3. We obtain 6-th generalized Lucas polynomial for k = 5, by using (2.3).   t1 it2 0 0 0 0  2i t1 it2 0 0 0    −t3   3 2 i t it 0 0 1 2 t2     det(+ C5,6 ) = det  4 −it3 4 −t23 i t it 0  1 2 t t 2 2   −it4 −t3  5 t54 i t1 it2    t2 t32 t22 t5 −it4 −t3 0 i t 1 t4 t3 t2 2

2

2

= t61 + 6t41 t2 + 9t21 t22 + 2t32 + 6t31 t3 + 3t23 + 12t1 t2 t3 +6t21 t4 + 6t2 t4 + 6t1 t5 = G5,6 (t). Theorem 2.4. Let k ≥ 2 and n ≥ 1 be integers, Gk,n the generalized Lucas polynomial and − Ck,n = (bij ) an n × n lower Hessenberg matrix,


given by

bij =

 −t2 ,      ti−j+1 (i−j) ,     

t2 ti−j+1 (i−j)

t2

if j = i + 1, if j ̸= 1 and 0 ≤ i − j < k,

.(i − j + 1),

if j = 1 and 0 ≤ i − j < k,

0,

otherwise

i.e.,



− Ck,n

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t1 2 3 tt23 2 .. .

       = tk  k k−1  t2   0   .  .. 0

−t2 t1 1 .. .

0 −t2 t1 .. .

tk−1 tk−2 2 tk tk−1 2

tk−2 tk−3 2 tk−1 tk−2 2

.. . 0

.. . 0

0 0 −t2 .. . .. .

··· ··· ···

tk−2 tk−3 2

··· .. . ···

.. . ···

···

0 0 0 .. .



        0    0    −t2  t1

where t0 = 1. Then, (2.4)

det(− Ck,n ) = Gk,n (t).

Proof. The proof is similar to the proof of Theorem 2.2, by using Lemma 2.1. □ Example 2.5. We obtain 5-th generalized Lucas polynomial for k = 4, by using (2.4).   t1 −t2 0 0 0  2 t1 −t2 0 0   t3   32 1 t −t 0  1 2 det(− C4,5 ) = det  t2  t3  4 t4 1 t1 −t2   t32  t22 t4 t3 0 1 t 1 t3 t2 2

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5t31 t2

= + = G4,5 (t).

+

5t1 t22

2

+ 5t21 t3 + 5t2 t3 + 5t1 t4

Corollary 2.6. If we rewrite equalities (2.3) and (2.4) for ti = 1 and k = 2, then we obtain det(+ Ck,n ) = det(− Ck,n ) = ln where ln are the ordinary Lucas numbers.

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Corollary 2.7. If we rewrite equalities (2.3) and (2.4) for t1 = 0, then we obtain det(+ Ck,n ) = det(− Ck,n ) = Rk,n (t) where Rk,n (t) are the generalized Perrin polynomials. Corollary 2.8. If we rewrite the right hand side of equalities (2.3) and (2.4) for t1 = x, tk = y, ti = 0 (2 ≤ i ≤ k − 1) and k = (p + 1), then we obtain det(+ Ck,n ) = det(− Ck,n ) = Lp,n (x, y) where Lp,n (x, y) are the generalized bivariate Lucas p-polynomials. Corollary 2.9. If we rewrite equalities (2.3) and (2.4) for t1 = 0 and ti = 1 (2 ≤ i ≤ k) and k = 3, then we obtain det(+ Ck,n ) = det(− Ck,n ) = rn where rn are the ordinary Perrin numbers. Now we show that determinants of Hessenberg matrices can be obtained by using combinations.

− Ck,n

and

+ Ck,n

Corollary 2.10.

∑ n ( |a| ) det(+ Ck,n ) = det(− Ck,n ) = ta1 . . . takk |a| a1,..., ak 1 a⊢n

Proof. It is obvious from Theorem 2.2, Theorem 2.4 and (1.1).

□

3. The permanent representations Let A = (ai,j ) be an n × n square matrix over a ring R. Then, it is well known that, the permanent of A is defined by n ∑ ∏ per(A) = ai,σ(i) σ∈Sn i=1

where Sn denotes the symmetric group on n letters. Lemma 3.1. [18] Let An be an n × n lower Hessenberg matrix for all n ≥ 1 and define per(A0 ) = 1. Then, per(A1 ) = a11 and for n ≥ 2 (3.1)

per(An ) = an,n per(An−1 ) +

n−1 ∑

n−1 ∏

r=1

j=r

(an,r

aj,j+1 per(Ar−1 )).


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We define two Hessenberg matrices − Hk,n and nents give the generalized Lucas polynomials.

+ Hk,n

whose perma-

Theorem 3.2. Let k ≥ 2 and n ≥ 1 be integers, Gk,n (t) the generalized Lucas polynomials and − Hk,n = (hrs ) an n × n lower Hessenberg matrix, given by  (r−s) . tr−s+1 ,  if s ̸= 1 and − 1 ≤ r − s < k, (r−s)  i  t2 t hrs = i(r−s) . r−s+1 if s = 1 and − 1 ≤ r − s < k, (r−s) .(r − s + 1),  t2   0, otherwise i.e., 

t1 2i 3i2 tt23 2 .. .

−it2 t1 i .. .

        t − Hk,n =  kik−1 tk ik−2 k−1 t2k−1 tk−2  2  tk 0 ik−1 k−1  t2   .. ..  . . 0 0 √ where t0 = 1 and i = −1. Then, (3.2)

0 −it2 t1 .. . tk−2 tk−3 2 t ik−2 k−1 tk−2 2

ik−3

.. . 0

0 0 −it2 .. .

··· ··· ···

tk−3 tk−4 2 t ik−3 k−2 tk−3 2

···

ik−4

.. . ···

··· .. . ···

0 0 0 .. . 0



           0     t1

per(− Hk,n ) = Gk,n (t).

Proof. The proof is similar to the proof of Theorem 2.2, by using Lemma 3.1. □ Example 3.3. We obtain the 3-rd generalized Lucas polynomial for k = 4, by using (3.2)   t1 −it2 0 t1 −it2  per(− H4,3 ) = per  2i −t3 3 t2 i t1 = t31 + 3t1 t2 + 3t3 . Theorem 3.4. Let k ≥ 2 and n ≥ 1 be integers, Gk,n (t) the generalized Lucas polynomials and + Hk,n = (pij ) an n × n lower Hessenberg matrix

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given by

pij =

   

ti−j+1

  

t2

(i−j)

,

if j ̸= 1 and − 1 ≤ i − j < k,

(i−j)

.(i − j + 1),

if j = 1 and 0 ≤ i − j < k,

t2 ti−j+1

0,

otherwise

i.e.,



+ Hk,n

       =       

t1 2 3 tt23 2 .. . tk k k−1 t2

0 0

t2 t1 1 .. .

0 t2 t1 .. .

0 0 t2 .. .

··· ··· ···

tk−1 t2k−2 tk t2k−1

tk−2 tk−3 2 tk−1 tk−2 2

tk−3 tk−4 2 tk−2 tk−3 2

···

.. . 0

.. . 0

.. . ···

··· .. . ···

0 0 0 .. . 0



           0     t1

where t0 = 1. Then, (3.3)

per(+ Hk,n ) = Gk,n (t).

Proof. The proof is similar to the proof of Theorem 2.2, by using Lemma 3.1. □ Corollary 3.5. If we rewrite equalities (3.2) and (3.3) for ti = 1 (1 ≤ i ≤ k) and k = 2, then we obtain per(− Hk,n ) = per(+ Hk,n ) = ln where ln are the ordinary Lucas numbers. Corollary 3.6. If we rewrite equalities (3.2) and (3.3) for t1 = 0 and ti = 1 (2 ≤ i ≤ k), then we obtain per(− Hk,n ) = per(+ Hk,n ) = Rk,n (t) where Rk,n (t) are the generalized Perrin polynomials. Corollary 3.7. If we rewrite the right hand side of equalities (3.2) and (3.3) for t1 = x, tk = y, ti = 0 (2 ≤ i ≤ k − 1) and k = (p + 1), then we obtain per(− Hk,n ) = per(+ Hk,n ) = Lp,n (x, y) where Lp,n (x, y) are the generalized bivariate Lucas p-polynomials.


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Corollary 3.8. If we rewrite equalities (3.2) and (3.3) for t1 = 0, ti = 1 (2 ≤ i ≤ k) and k = 3, then we obtain per(− Hk,n ) = per(+ Hk,n ) = rn where rn are the ordinary Perrin numbers. Now we show that permanent of Hessenberg matrices + Hk,n can be obtained by using combinations. Corollary 3.9. per(− Hk,n ) = per(+ Hk,n ) =

− Hk,n

and

∑ n ( |a| ) ta1 . . . takk |a| a1,..., ak 1 a⊢n

Proof. It is obvious from Theorem 3.2, Theorem 3.4 and (1.1).

□

4. Determinant and permanent of a Hessenberg matrix Gibson [3] gave an identity between permanent and determinant of a semitriangular matrix. We give a different proof of this identity for Hessenberg matrices. Theorem 4.1. Let − An

+ An

be the Hessenberg matrix in (2.1) and

= (bij ) an n × n Hessenberg matrix, given by  if j − i > 1,  0, −aij , if j − i = 1, bij =  aij , otherwise

i.e.,



a11 a21 a31 .. .

−a12 a22 a32 .. .

0 −a23 a33 .. .

··· ··· ···

    A =  − n    an−1,1 an−1,2 an−1,3 · · · an,1 an,2 an,3 · · ·

0 0 0 .. . −an−1,n an,n

     .   

Then, (4.1)

det(− An ) = per(+ An ) and det(+ An ) = per(− An ).

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Proof. To prove (4.1), we use the mathematical induction on m. The result is true for m = 1 by hypothesis. Assume that it is true for all positive integers less than or equal to m, namely det(− Am ) =per(+ Am ). Then, by using (2.2) and (3.1), we have det(− Am+1 ) = am+1,m+1 det(− Am ) m m ∏ ∑ bj,j+1 det(− Ar−1 )] + [(−1)m+1−r am+1,r r=1

j=r

= am+1,m+1 per(+ Am ) m m ∑ ∏ m+1−r + (−aj,j+1 )per(+ Ar−1 )] [(−1) am+1,r r=1

j=r

= am+1,m+1 per(+ Am ) m m ∑ ∏ m+1−r m+1−r + [(−1) am+1,r (−1) aj,j+1 per(+ Ar−1 )] r=1

j=r

= am+1,m+1 per(+ Am ) +

m ∑

[am+1,r

r=1

m+1 ∏

aj,j+1 per(+ Ar−1 )]

j=r

= per(+ Am+1 ). Therefore, the result is true for all positive integers.

□

Conclusion Generalized Lucas polynomials are a general form of several polynomials and number sequences. Therefore any result obtained from these polynomials is applicable to the others. In addition, the relation between determinant and permanent of a Hessenberg matrix make it possible to transfer any result between them. Acknowledgments The authors would like to express their appreciation to the anonymous reviewers for their careful reading and making some useful comments, improving the presentation of the paper.


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K. Kaygısız Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpa¸sa University, 60250 Tokat, Turkey Email: [email protected] A. S ¸ ahin Department of Mathematics, Faculty of Education, Gaziosmanpa¸sa University, 60250 Tokat, Turkey Email: [email protected]