Yazar "Yilmaz, Semih" seçeneğine göre listele

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    Complex Factorization By Chebysev Polynomials
    (Univ Studi Catania, Dipt Matematica, 2018) Sahin, Murat; Tan, Elif; Yilmaz, Semih
    Let {a(i)}, {b(i)} be real numbers for 0 <= i <= r - 1, and define a r-periodic sequence {v(n)} with initial conditions v(0) , v(1) and recurrences v(n) = a(t)v(n-1) vertical bar b(t)v(n-)(2) where n t (mod r) (n >= 2). In this paper, by aid of Chebyshev polynomials, we introduce a new method to obtain the complex factorization of the sequence {v(n)} so that we extend some recent results and solve some open problems. Also, we provide new results by obtaining the binomial sum for the sequence {v(n)} by using Chebyshev polynomials.
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    The eigenvalues of r-periodic tridiagonal matrices by factorization of some recursive sequences
    (Elsevier Science Bv, 2018) Sahin, Murat; Yilmaz, Semih
    We introduce r-periodic tridiagonal matrices for given integer r >= 2. In which the entries on the principle diagonal can be any r-periodic sequence. If the entries on the principle diagonal are equal then the calculation of the eigenvalues of corresponding tridiagonal matrices is relatively easy, but when the entries are not equal, the calculation becomes much more difficult. So, some explicit formulas could be given only for the eigenvalues of certain types of the 2-periodic tridiagonal matrices so far. We give a new algorithm to find the eigenvalues of certain r-periodic tridiagonal matrices and give some results by implementing it in a symbolic programming language. Our algorithm also finds the zeros of some families of polynomials with integer coefficients. The degree of these polynomials can be chosen very high. Also, we give some new properties of the generalized continuant and then we solve an open problem by determining a complex factorization of certain r-periodic sequences. Finally, we generalize existing results by giving the explicit formulas for the eigenvalues of some r-periodic tridiagonal matrices for r = 2, 3 and 4. (C) 2017 Elsevier B.V. All rights reserved.
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    A note on bi-periodic Fibonacci and Lucas quaternions
    (Pergamon-Elsevier Science Ltd, 2016) Tan, Elif; Yilmaz, Semih; Sahin, Murat
    Motivated by the our recent work in Tan et al., 2016, related to the bi-periodic Fibonacci quaternions, here we introduce the bi-periodic Lucas quaternions that gives the Lucas quaternions as a special case. We give the generating function and the Binet formula for these quaternions. Also, we give the relationships between bi-periodic Fibonacci quaternions and bi-periodic Lucas quaternions. (C) 2016 Elsevier Ltd. All rights reserved.
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    On a new generalization of Fibonacci quaternions
    (Pergamon-Elsevier Science Ltd, 2016) Tan, Elif; Yilmaz, Semih; Sahina, Murat
    In this paper, we present a new generalization of the Fibonacci quaternions that are emerged as a generalization of the best known quaternions in the literature, such as classical Fibonacci quaternions, Pell quaternions, k-Fibonacci quaternions. We give the generating function and the Binet formula for these quaternions. By using the Billet formula, we obtain some well-known results. Also, we correct some results in [3] and [4] which have been overlooked that the quaternion multiplication is non commutative. (C) 2015 Elsevier Ltd. All rights reserved.