On Generating Tridiagonal Matrices of Generalized ( s, t ) -Pell, ( s, t ) -Pell Lucas and ( s, t ) -Modiﬁed Pell Sequences

In this study, we deﬁne some tridigional matrices depending on two real parameters. By using the determinant of these matrices, the elements of ( s, t )-Pell, ( s, t )-Pell Lucas and ( s, t )-modiﬁed Pell sequences with even or odd indices are generated. Then we construct the inverse matrices of these tridigional matrices. We also investigate eigenvalues of these matrices.


Introduction and Preliminaries
Special integer sequences are encountered in different branches of science, art, nature, the structure of our body. So it is a popular topic in applied mathematics. Two of the special integer sequences are the Pell and Pell Lucas sequences. By changing the initial conditions but preserving the recurrence relation the Pell Lucas sequence is obtained. The authors investigated some sum formulas for Pell numbers in [2]. Gulec and Taskara generalized the Pell and Pell Lucas numbers by using two parameters in [3]. The authors generalized the modified Pell sequence similarly in [4]. By the determinant of the tridiagonal matrix, the values of the Fibonacci and Lucas numbers are demonstrated in [8]. Feng gave Fibonacci identities via determinant of the special tridiagonal ℘ n (s, t) = 2s℘ n−1 (s, t) + t℘ n−2 (s, t), ℘ 0 (s, t) = 0, ℘ 1 (s, t) = 1, n (s, t) = 2s n−1 (s, t) + t n−2 (s, t), 0 (s, t) = 2, 1 (s, t) = 2s, ℵ n (s, t) = 2sℵ n−1 (s, t) + tℵ n−2 (s, t), ℵ 0 (s, t) = 1, ℵ 1 (s, t) = s, for n ≥ 2 in [3,4].
Special integer sequences are obtained with special numerical choices for (s, t)-Pell and (s, t)-Pell Lucas, (s, t)-modified Pell sequences. For example, if s = t = 1, then we get well-known Pell, Pell Lucas, modified Pell sequences. If s = 1, t = k, then we get k-Pell, k-Pell Lucas, and modified k-Pell sequences.
Let us consider a tridiagonal matrix as The inverse of a matrix A can be obtained by the formula A −1 = (cof (A)) T det A , where (cof (A)) T is the transpose of the cofactor matrix A or adjugate matrix of A [4]. Let T be a nonsingular tridiagonal matrix as 232 S. Uygun Usmani [6] gave a formula for the inverse of this matrix T −1 = (t i,j ) as ...n, with the initial conditions θ 0 = 1 and θ 1 = a 1 . Observe that θ n = det (T ) .
with the initial conditions φ n+1 = 1 and φ n = a n .
1.1 Some properties of tridiagonal matrices A p,n by (s, t)-Pell sequence Theorem 1. Assume that A p,n is an n × n tridiagonal matrix defined as Then the determinant of A p,n is Proof. The proof is made by induction applied on n. For n = 1, we have det(A p,1 ) = ℘ 2 = 2s. Assume that det(A p,n−1 ) = ℘ n , det(A p,n ) = ℘ n+1 for n > 2. Then (s, t)-Pell sequence is also obtained by using the following tridiagonal matrix with complex entries. Assume that A n is an n × n matrix defined as

S. Uygun
Then it is easily seen that the determinant of A n is also (n + 1)th element of the (s, t)-Pell sequence det(A n ) = ℘ n+1 .
For the inverse of A p,n , by using (2), it is obtained that Therefore the inverse of A p,n is displayed by The elements of the cofactor matrix are given as It is well-known that |cof (A p,n )| = |adj(A p,n )| = |A p,n | n−1 = ℘ n−1 n+1 . By using cofactor matrix, we get some properties of (s, t)-Pell sequence. For n = 2, we get For n = 4, we get Eigenvalues of the matrices A p,n construct the spectra of the A p,n . By using the property (3), the sequence of the spectra of A p,n for n = 1, 2, 3, 5 is , the sequence of the spectra of the matrices A p,n for n = 2, 3, 4, 5, 6 is computed by using the Matlab Program as Evidently, the product of eigenvalues is the determinant of the matrix and the sum of eigenvalues is the trace of the matrix. Therefore (2s + √ 2i cos( πj n + 1 )).

Some properties of tridiagonal matrices E p,n by even (s, t)-Pell sequence
Assume that E p,n (p) is an n × n tridiagonal matrix defined as Then the determinant of E p,n is computed by (1) as For the inverse of E p,n , the values in (2) are computed as a 1 = 2s, Therefore the inverse of E p,n is displayed as If all entries of the matrix are real and nonnegative, then the matrix is called positive. All eigenvalues are real if the matrix positive and tridiagonal [4].
Therefore all eigenvalues of E p,n are real if s, t ≥ 0. If we choose s = t = 1, then the sequence of the spectra of the matrix E p,n for n = 2, 3, 4, 5, 6 is given in the following result with the help of the Matlab program Evidently λ i = tr(E p,n ) = (n − 1)(4s 2 + 2t) + 2s and λ i = det(E p,n ) = ℘ 2n .
If we take care of the spectra, one of the eigenvalues is 2 = 2s if s = t = 1 for all positive integer n. And the minimum eigenvalue of spectra converges to 2 = 2s, the maximum eigenvalue of spectra converges to 4s 2 + 4t.

The properties of tridiagonal matrices O p,n by odd (s, t)-Pell sequence
Assume that O p,n is an n × n tridiagonal matrix defined as Then the determinant of O p,n is given by (1) as det O p,n = ℘ 2n+1 .

S. Uygun
For the inverse of O p,n , the values are computed by (2) as Therefore the inverse of O p,n is given by Matrices O p,n are symmetric so the eigenvalues are real. If s = t = 1, the sequence of the spectra of the matrices O p,n for n = 2, 3, 4, 5, 6 is given in the following: Then If we take care of the spectra, minimum eigenvalue converges to 4s 2 . The maximum eigenvalue of spectra converges to 4s 2 + 4t.
Theorem 2. If λ i is an eigenvalue of the matrix O p,n , then λ i + 8s + 4 is an eigenvalue of O p,n (s + 1).
Proof. λ i is an eigenvalue of O p,n , then

Some properties of tridiagonal matrices A Q,n by (s, t)-Pell
Lucas sequence Assume that A Q,n is an n × n tridiagonal matrix defined as Then the determinant of A Q,n det A Q,n = n .

S. Uygun
(s, t)-Pell Lucas sequence is also obtained by using the following symmetric matrix with complex entries. Assume that A n is an n × n tridiagonal matrix defined as Then the determinant of A n is given by (1) as The sequence of the spectra of the matrices A Q,n for n = 2, 3, 4, 5, 6 is given in the following: Evidently, λ i = tr(A Q,n ) = 2sn and λ i = det(A Q,n ) = n . For the inverse of A Q,n , by using (2), it is obtained that i > 1 Therefore the inverse of A Q,n is the following matrix

Some properties of tridiagonal matrices E Q,n by even (s, t)-Pell Lucas sequence
Assume that E Q,n is a n × n tridiagonal matrix defined as Then the determinant of E Q,n is given by (1) det E Q,n = 2n .
For the inverse of E Q,n , the values are computed as, The sequence of the spectra of the matrices E Q,n for n = 2, 3, 4, 5, 6, are given in the following Then, λ i = tr(E Q,n ) = n(4s 2 + 2t) and λ i = det(E Q,n ) = 2n .
If we take care of the spectra, minimum eigenvalue converges to 4s 2 . The maximum eigenvalue of spectra converges to 4s 2 + 4t.

Some properties of tridiagonal matrices O Q,n by odd (s, t)-Pell
Lucas sequence Assume that O Q,n is an n × n tridiagonal matrix defined as If s = t = 1, then the sequence of the spectra of the matrices O Q,n for n = 2, 3, 4, 5, 6 is given in the following The sequence of maximum eigenvalue is increasing and converges to 4s 2 +4t = 8. So we can say lim

Some properties of tridiagonal matrices A q,n by (s, t)-Modified Pell sequence
Assume that A q,n is an n × n tridiagonal matrix defined as

S. Uygun
(s, t)-modified Pell sequence is also obtained by using the following symmetric matrix with complex entries. Assume that A n is an n × n tridiagonal matrix defined as Then the determinant of A n is given by (1) as The sequence of the spectra of the matrices A q,n for n = 2, 3, 4, 5, 6 is given in the following: For the inverse of A q,n , by using (2), it is obtained that Therefore, the inverse of A q,n is the following matrix Assume that E q,n is an n × n tridiagonal matrix defined as Then the determinant of E q,n is given by (1) det E q,n = ℵ 2n .
Therefore, we get The sequence of the spectra of the matrices E q,n for n = 2, 3, 4, 5, 6, are given in the following Clearly, λ i = tr(E q,n ) = n(4s 2 + 2t) − 2s 2 − t and λ i = det(E q,n ) = ℵ 2n . If we take care of the spectra, minimum eigenvalue converges to 2s 2 . The maximum eigenvalue of spectra converges to 4s 2 + 4t.