Factorizations of the Negatively Subscripted Balancing and Lucas-Balancing Numbers

with B1 = 1, B2 = 6 [1]. Also in [1], it is shown that, if x is a balancing number, then 8x +1 is a perfect square. If x is balancing number then the positive square root of 8x + 1 is called a Lucas-balancing number denoted by Cn [8]. Observe that C1 = 3, C2 = 17 and the Lucas-balancing numbers Cn satisfy the recurrence relation Cn+1 = 6Cn − Cn−1, n ≥ 2, (2) identical with that for balancing numbers. By using the formulas (1) and (2), we can extend these sequences backward, to get


Introduction
It is well-known that the sequence of balancing numbers {B n } are solutions of the recurrence relation, for n ≥ 2, with B 1 = 1, B 2 = 6 [1].Also in [1], it is shown that, if x is a balancing number, then 8x 2 + 1 is a perfect square.If x is balancing number then the positive square root of 8x 2 + 1 is called a Lucas-balancing number denoted by C n [8].Observe that C 1 = 3, C 2 = 17 and the Lucas-balancing numbers C n satisfy the recurrence relation identical with that for balancing numbers.By using the formulas (1) and (2), we can extend these sequences backward, to get 2000 Mathematics Subject Classification: 11 B 39, 11 B 83 162 Prasanta Kumar Ray In [8], Panda has shown that, the Lucas-balancing numbers are associated with balancing numbers in the way Lucas numbers are associated with Fibonacci numbers.In [7], Panda and Ray have proved that the Lucas-balancing numbers are nothing but the even ordered terms of the associated Pell sequence.Also they have shown that the n th balancing numbers are product of n th Pell numbers and n th associated Pell numbers.In [3], K. Liptai searched for those balancing numbers which are Fibonacci numbers too.He proved that the only Fibonacci number in the sequence of balancing numbers is 1.In a similar manner, in [4], he proved that there are no Lucas numbers in the sequence of balancing numbers.L. Szalay in [5] got the same result.In [9], by using Chebyshev polynomials of first and second kind, Ray has obtained nice product formulae for both balancing and Lucas-balancing numbers.
In this paper, we consider negatively subscripted balancing and Lucas-balancing numbers and find some tridiagonal matrices whose determinant and permanent equal to these numbers.In the final section of the paper, we give the factorization of these numbers by using the first and second kinds of Chebyshev polynomials.

Negatively Subscripted Balancing and Lucas-balancing Numbers
In this section, we define some tridiagonal matrices and then prove that the determinant and permanent of these matrices are equal to the negatively subscripted balancing and Lucas-balancing numbers.For simplicity, we present some known definitions which will be used subsequently.Definition 2.1.[2,6].If A = (a ij ) is a square matrix of order n, then the permanent of A, denoted by perA is defined by where the summation extends over all permutations σ of the symmetric group S n .
There are many applications of permanent that are given in [6].
) is an m × n matrix with row vectors r 1 , r 2 , . . .r m , then A is called contractible on column (resp.row) k if column (resp.row) k contains exactly two nonzero entries.Suppose A is contractible on column k with a ik = 0 = a jk and i = j.Then the matrix A ij: k of order (m − 1) × (n − 1) can be obtained from A by replacing row i with a jk r i + a ik r j and deleting row j and column k is called the contraction of A on column k relative to rows i and j.If A is contractible on row k with a ki = 0 = a kj and i = j, then the matrix T is called the contraction of A on row k relative to columns i and j.
Definition 2.3.[2].A matrix A is called convertible if there exists an n × n (1, -1)-matrix H such that det(A • H) = perA, where A • H is well known Hadamard product of A and H.We call H is a converter of A.
In this paper, every contraction will be on the first column using the first and second rows.It is well known that, if A be a nonnegative integral matrix of order n > 1 and let B be a contraction of A. Then, perA = perB. ( First, we start with negatively subscripted balancing numbers.We introduce the sequence of matrices , and the following theorem holds.
Theorem 2.4.If the sequence of tridiagonal matrices where B −n is the n th negatively subscripted balancing number.
Proof: Clearly for n = 1, n = 2, we have Let the p th contraction of M n be M p n , where 1 ≤ p ≤ n − 2. Using Definition 2.2, Prasanta Kumar Ray the matrix M n can be contracted on Column 1 as .
Again the matrix M 1 n can be contracted on Column 1 as, Proceeding in this way, we obtain, for 3 ≤ r ≤ n − 4, , and therefore Using (3), we get which ends the proof of the theorem.✷ For negatively subscripted Lucas-balancing numbers, consider another n × n tridiagonal matrix We prove the following theorem.
Theorem 2.5.If the sequence of tridiagonal matrices where C −n is the n th negatively subscripted Lucas-balancing number.
Proof: The theorem holds for n = 1, n = 2, because Let D p n be the p th contraction of D n where 1 ≤ p ≤ n − 2. By virtue of Definition 2.2, the matrix D n can be contracted on Column 1 as The matrix D 1 n can be contracted on Column 1 as, Continuing in this way, we obtain, for 3 ≤ r ≤ n − 4, , and therefore By contraction of D n−3 n on Column 1, gives Again by using (3), we get This completes the proof of the theorem.✷ By virtue of Definition 2.3, we need a suitable matrix H for Hadamard product.Since H is an n × n (-1, 1)-matrix, we can write .

Prasanta Kumar Ray
Let the Hadamard products M n •H and D n •H respectively denoted by the matrices P n and Q n be given by It is well known that the value of the following determinant, is independent of x (see p.105, [10]).
Therefore, using the above result and considering the following matrices

Factorization of Negatively Subscripted Balancing and Lucas-balancing Numbers
In this section we find the eigenvalues of the two tridiagonal matrices whose determinants are associated with the negatively subscripted balancing and Lucas-balancing numbers.Then with the help of Chebyshev polynomials of first and second kind, we also obtain the factorization of these numbers.Theorem 3.1.If B −n is the n th negatively subscripted balancing number, then for n ≥ 1 This shows that −6 + α k , 1 ≤ k ≤ n, are eigenvalues of Pn .Thus for n ≥ 1, we have Recall that each α k is a root of the characteristic polynomial p(α) = det(R n − αI), and since R n − αI is a tridiagonal matrix, that is, , we get the following recursive formulas for the characteristic polynomials: This family of polynomials can be transformed into another family {U n (x), n ≥ 1} by the transformation α = −2x to get, Observe that the family {U n (x), n ≥ 1} is a set of Chebyshev polynomials of second kind.It is well known that for x = cos θ, the Chebyshev polynomials of the second kind can be written as which when equal to zero gives θ k = πk n+1 , k = 1, 2, • • • , n.Therefore, we get The transformation α = −2x, gives the eigenvalues of R n as Thus from ( 4), ( 6) and ( 7), we get the desired result as Proof: From equation (5), we have det Qn = (−1) n−2 C −n .If e j is the j th column of the identity matrix I, we observe that det(I + e 1 e T 1 ) = 2. Thus we may write Factorizations of the Negatively Subscripted Balancing 171 Also observe that the right hand side of ( 8) can be expressed as with corresponding eigenvectors Y k , then for each j, This shows that the eigenvalues of the matrix In order to find γ ′ k s, we recall that each γ k is a zero of the characteristic polynomial .
Using the transformation γ = −2x, the family of the above polynomial can be transformed to a new family {T n (x), n ≥ 1} where, Again observe that the family {T n (x), n ≥ 1} is a set of Chebyshev polynomials of first kind.It is well known that, for x = cos θ, the Chebyshev polynomials of the first kind can be written as T n (x) = cos nθ, which when equal to zero gives θ k = π(2k−1)

Factorizations
of the Negatively Subscripted Balancing 165 By contraction of M n−3 n on Column 1, gives