# Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz matrix and the Chebyshev polynomials

### Ahmet Zahid Küçük

,### Murat Düz

Journal of Applied Mathematics and Computational Mechanics |
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RELATIONSHIPS BETWEEN THE PERMANENTS OF A
CERTAIN TYPE OF *k*-TRIDIAGONAL SYMMETRIC TOEPLITZ MATRIX AND THE CHEBYSHEV
POLYNOMIALS

Ahmet Zahid Küçük^{ }^{1}, Murat Düz^{ }^{2}

^{1} Eskipazar
Vocational School

Turkey

^{2} Department of Mathematics Faculty of Science, Karabuk University

Turkey

azahidkucuk@karabuk.edu.tr, mduz@karabuk.edu.tr

Received: 16 December 2016; accepted:
13 March 2017

**Abstract.** In this
study, the recursive relations between the permanents of a certain type
of the *k*-tridiagonal symmetric Toeplitz matrix with complex entries and
the Chebyshev polynomials of the second kind are presented.

*MSC 2010:** 33C45*

*Keywords: **permanent, Toeplitz matrix,
k-tridiagonal matrix, Chebyshev polynomials, recurrence relation*

1. Introduction

A *k*-tridiagonal symmetric Toeplitz
matrix can be written in the form

where

. | (1) |

The Toeplitz structured tridiagonal matrix family can be seen with many studies in different areas. Some of them include the solution of ordinary and partial differential equations [1] in the applied mathematics, including the equilibrium problem [2] in the statistical physic, wireless and sensor network applications [3, 4] about control theory in computer sciences and the sensitivity of MRNA [5] in molecular biology.

The aim of this study is to point out some
relationships between the permanents of the certain type of *k*-tridiagonal
symmetric Toeplitz matrix and the Chebyshev polynomials of the second kind. The
definition of the *k*-tridiagonal symmetric
Toeplitz matrix that is used in this study is given below.

**Definition:** Let the *k*-tridiagonal
symmetric Toeplitz matrix of order with
first row be

(2) |

, and are positive integers such that

The matrix was
pronounced as the tridiagonal instead of the *k*-tridiagonal throughout in
this study. Also, it is worth noting that if and then the and matrices convert a pentadiagonal and
heptadiagonal matrix respectively. But, this nomenclature for was not used in this study.

In the literature, much research has been gathered
on the relationships between the permanent or determinant of some tridiagonal
or Toeplitz matrices and the
famous number sequences or the linear recurrences. For example, Minc [6] gave
a recursive formula for the permanent of a general tridiagonal Toeplitz matrix.
Lehmer [7] demonstrated some generalizations for the permanent of a tridiagonal
matrix by using expansion by minors. Kılıc and Tascı [8]
presented some relationships between the permanents of some tridiagonal
matrices and some famous number sequences. Jina and Trojovsky [9] presented new
results about the relationships between the permanents of some tridiagonal
matrices and the Fibonacci numbers. Matousova and Trojovsky [10] showed a
generalization for sequences of tridiagonal matrices which is related to the
sequence of Fibonacci numbers. Kaygısız and Sahin [11] obtained the
permanents of some tridiagonal matrices with complex elements in terms of k
sequences of generalized order-k Fibonacci numbers.
Kırklar and Yılmaz [12] gave an important representation for the
permanent of
a *k*-tridiagonal *k*-Toeplitz matrix by using the direct sum of the
matrices. Yılmaz and Bozkurt [13] gave one type of tridiagonal matrix
whose permanents are Jacobsthal numbers. Of course, the studies on the
determinants of special matrices take more space than on the permanents of
these matrices. We focused on the few that are closest to our study. Elouafi [14]
gave the formulas which involve determinants whose entries are the Chebyshev
polynomials for a banded symmetric
Toeplitz matrix. Cahill et al. [15] showed some special tridiagonal matrices
whose determinant gives the Chebyshev polynomials. Kılıç [16]
presented the two different representations depending on the parity of *n*
for the permanent of general
2-tridiagonal Toeplitz matrix that emblematized this matrix with H(a,b,-c).
Asci, Tasci and Al-Mikkawy [17] gave algorithms for permanents of *k*-tridiagonal
matrices using LU factorization. In Trench [18] gave a recursive relation for
the eigenvalues of a 2-tridiagonal Toeplitz matrix. Bergun and Hoggatt [19]
gave the recursive relations among the determinants of a general tridiagonal,
2-tridiagonal and 3-tridiagonal Toeplitz matrix and also gave a representation
for the determinant of the general *k*-tridiagonal Toeplitz matrix with
respect to the determinant of the tridiagonal Toeplitz matrix. Our study
focuses on the relationships between the permanents of a certain type of *k*-tridiagonal
symmetric Toeplitz matrix with complex entries and the Chebyshev polynomials of
the second kind.

Chebyshev polynomials being a useful
mathematical tool, are often utilized
in the mathematics as well as other fundamental science and the engineering.
Chebyshev polynomials of the second kind is a polynomial of degree *n* in *x*
defined in [20] by

where . The family of the Chebyshev polynomial of the second kind satisfies the recurrence relation

(3) |

where , , . We have referred to Chebyshev
polynomials of the second kind as Chebyshev *U* polynomials throughout in
this study. The first few Chebyshev *U* polynomials are

(4) |

Let be the symmetric group which consist of all permutations of and be the element of this group where . Then, the permanent of an square matrix is defined by

(5) |

The permanent is a function of the matrix
elements such as the determinant without using signs for the permutations. For
this reason, some operations that
are applied on the determinant can also be used on the permanent. The Laplace
expansion is one of these operations. We note that all of the signs used in the
Laplace expansion of minors are positive. It should be noted that we used the
Laplace expansion for the permanent throughout in this study. Despite gaining
more prominence in the recent studies on the permanent such as in [12], the
*contraction* method, which is given by Brualdi and Gibson [5], doesn’t work
for the permanent of a complex matrix.

2. Main results

2.1. Relationships between
and the Chebyshev *U*
polynomial

**Theorem 1: **The
permanent of the matrix is

(6) |

where is the *n*th Chebyshev *U*
polynomials and .

**Proof:** We will
use mathematical induction method in this proof.

For

Thus (6) is provided.
Assume that the equality (6) holds for *n*. So, we have

(7) |

We will now show that equality (6) holds for , namely

. | (8) |

Firstly, we expand the by using the Laplace expansion according to the first row of . Then, we obtain

. |

Secondly, we expand the permanent which appears in the second term of the right hand side of the above equation with respect to its first column. So, we obtain

(9) |

which is desired.

**Theorem 2:** The
permanent of the matrix is

(10) |

where is the *n*th Chebyshev *U* polynomials and .

**Proof: **We will
use the mathematical induction method for the proof.

Suppose that . Then

Thus, the equality (10) is true for . Assume that
the equality (10) holds for *n*, i.e.

. | (11) |

We will now show that the equality (10) is also true for , namely

. | (12) |

Let us write by using the Laplace expansion with respect to its first column. Then

(13) |

Let’s expand the permanent that appears in the second term of right hand side of the equality (13) with respect to its first row. So, we have:

(14) |

If the permanent appears in the second term of right hand side of the (14) is expanded with respect to the its first row, then will be equal to

(15) |

Lastly, we expand the permanent that appears in the third term of right hand side of the (15) with respect to its first column. So, we obtain

. | (16) |

We rewrite the equality (16) by using (11),

(17) |

We have two cases depending on the parity of *n*.
Let *n* be *odd*. If we arrange
the equality (17), then we get the following result (18).

(18) |

It is easy to see that the result (18) is
obtained also if *n* is *even*. Thus, the proof is completed.

**Theorem 3:** There is a representation for the
Chebyshev *U* polynomials in terms of the permanent of as following:

(19) |

where for . |

**Proof:** Using
Theorem 2, we write that

. | (20) |

If we replace the upper limit of the first summation in the equality (20) with So, the equality (20) will take the form of

(21) |

such that the right side of (21) is equal to .

**Conjecture 4:**
For

(22) |

and for

(23) |

where is the *n*th
Chebyshev *U* polynomial.

Because of that becomes difficult obtaining of
the representations with respect
to the Chebyshev *U* polynomials
for each values of , the study will continue
the following section.

2.2. The Recursive Formulas for

In accordance with the Theorem 1, it can be
seen that the defining recurrence of the Chebyshev *U* polynomials, given
by (3), is true for .

**Result 5:** For

(24)

where , .

Hereby, we should emphasize that also the equality (16), which is a natural result of the Theorem 2, is actually a recursive formula as like (24). This is stated below.

**Result 6:** For

with initial conditions , , and .

**Proof:** If the equalities (6) and (19) uses
together in the equality (24) then
the expression which is numbered in the first section with (16) can be
obtained.

**Theorem 7:** If is integer then

(25) |

is true such that , .

**Proof:** It is easily proven using the induction
method by .

Firstly, we must demonstrate the equality (25) is true for , i.e.

. | (26) |

By using the Theorem 2, the left side of the equality (26) is which yields to . Likewise, the right side of the equality (26) is which also yields to . Thus, it is easy to see that the both sides of (26) are equal to each other.

Now, assume that the equality (26) holds for , i.e.

(27) |

where . We will show that the equality (26) also holds for . If equality (16) is rewritten by putting instead of , then

(28) |

is obtained. If the equality (27) into consideration in the equality (28), then

is obtained which is desired.

Now let us demonstrate the generalization we obtained for .

**Conjecture 8:**
If then

(29) |

where and .

In examinating the Conjecture 8, firstly the value of is assignment, then the values of are assingment.

**Remark 9:** According to the Conjecture 8, it can be
clearly stated that ,… cannot be written in terms
of themselves like the recurrence (16) which is emphasized as the Result 6.

Let’s explain this remark by giving an example. In the case of , becomes equal to. So, it can be obtained following equalities for the first few values of :

, | (30) |

, | (31) |

, | (32) |

. | (33) |

None of the terms of these equations (30), (31), (32), (33) is repeated in another equation. So, it's the evidence that the recursive formula which is written for in terms of itself and is provided for all consecutive values cannot be obtained.

However, a recursive representation for given below was obtained.

**Conjecture 10: **Let and be the positive integers and . Then

(34) |

where .

**Result 11:**
Taking into account the Conjecture 8 and the Conjecture 10 we get

(35) |

where and is a positive integer. If the equality (35) is rearranged by taking
into the Theorem 1 consideration, we obtain the representation of Chebyshev *U*
polynomials in terms of the permanent of as
follows:

(36) |

where and is the positive integer and is the rth Chebyshev *U* polynomials.

As an example of using the equality (35), let’s take the :

(37) |

So, we get respectively for .

, , . |

3. Conclusions

In this study, it can be seen that the
Chebyshev polynomials can be expressed
in terms of the permanents of tridiagonal *k*-Toeplitz matrices which are both band and circulant matrices.

Additionally, some recursive relations for
the permanent of the certain type
of *k*-tridiagonal symmetric Toeplitz matrix as conjectures are given.

It can be seen that the recursive representations obtained in this study are similar to the results in the [19].

As a further study, the representation of with respect to the Cheby- shev polynomials can be generalized for in the Section 2.1.

References

[1] Mazilu I., Mazilu D.A., Williams H.T., Applications of tridiagonal matrices in non-equilibrium statistical physics, Electronic Journal of Linear Algebra 2012, 24, 7-17.

[2] Fontanelli D., Palopoli L., Passerone R., On the Global Convergence of a Class of Distributed Algorithms for Maximizing the Coverage of a WSN, 28th Chinese Control Conference, Shanghai 2009 (IEEE, 2009), 7885-7890.

[3] Martínez S., Bullo F., Cortés J., Frazzoli E., On synchronous robotic networks - Part I: Models, tasks, and complexity, IEEE Transactions On Automatic Control 2007, 52(12).

[4] Poker G., Margaliot M., Tuller T., Sensitivity of mRNA translation, Scientific Reports 5, Article number: 12795, (2015).

[5] Brualdi R.A., Gibson P.M., Convex polyhedra of doubly stochastic matrices. I: Applications of the permanent function, The Journal of Combinatorial Theory A 1977, 22, 194-230.

[6] Minc H., On permanents of circulants, Pacific Journal of Mathematics 1972, 42(2).

[7] Lehmer D., Fibonacci and related sequences in periodic tridiagonal matrices, Fib. Quart. 1975, 13, 150-158.

[8] Kilic E., Tasci D., On the permanents of some tridiagonal matrices with applications to the Fibonacci and Lucas numbers, Rocky Mountain J. Math. 2007, 37(6), 1953-1969.

[9] Jina J., Trojovsky P., On permanents of some tridiagonal matrices connected with Fibonacci numbers, International Journal of Pure and Applied Mathematics 2014, 97(1), 79-87.

[10] Matousova I., Trojovsky P., On a sequence of tridiagonal matrices, whose permanents are related to Fibonacci and Lucas numbers, International Journal of Pure and Applied Mathematics 2015, 105(4), 715-723.

[11] Kaygısız K., Sahin A., Determinant and permanent of Hessenberg matrix and Fibonacci type Numbers, Gen. Math. Notes 2012, 9(2), 32-41.

[12] Kırklar E., Yılmaz F., A note on k-tridiagonal k-Toeplitz matrices, Alabama Journal of Mathematics 2015, 39.

[13] Yılmaz F., Bozkurt D., The adjacency matrix of one type of directed graph and the Jacobsthal numbers and their determinantal representation, Journal of Applied Mathematics 2012.

[14] Elouafi M., On a relationship between Chebyshev polynomials and Toeplitz determinants, Applied Mathematics and Computation 2014, 229, 27-33.

[15] Cahill N.D., Derrico J.R., Spence J.P., Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quarterly 2003, 41(1), 13-19.

[16] Kılıç E., On a constant-diagonals matrix, Applied Mathematics and Computation 2008, 204(1), 184-190.

[17] Asci M., Tasci D., El-Mikkawy M., On determinants and permanents of k-tridiagonal Toeplitz matrices, Util. Math. 2012, 89, 97-106.

[18] Trench W.F., Banded Symmetric Toeplitz Matrices (Lecture of Mathematic Seminar, Trinity University, 2009).

[19] Bergun G.E., Hoggatt V.E., A family of tridiagonal matrices, Fibonacci Quart. 1978, 16, 285-288.

[20] Mason J.C., Handscomb D.C., Chebyshev Polynomials, Chapman Hall/CRC, Boca Raton 2003.