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    On bounds for the norms of circulant matrices with the generalized Fibonacci and Lucas numbers
    (Charles Babbage Res Ctr, 2011) Sozer, Musa; Ipek, Ahmet; Kilicoglu, Oguz
    This paper is an extension of the work [On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. and Comp., 160 (2005), 125-132.], in which for some norms of the circulant matrices with classical Fibonacci and Lucas numbers it is obtained the lower and upper bounds. In this new paper, we generalize the results of that work.
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    On Pell Quaternions and Pell-Lucas Quaternions
    (Springer Basel Ag, 2016) Cimen, Cennet Bolat; Ipek, Ahmet
    The main object of this paper is to present a systematic investigation of new classes of quaternion numbers associated with the familiar Pell and Pell-Lucas numbers. The various results obtained here for these classes of quaternion numbers include recurrence relations, summation formulas and Binet's formulas.
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    On the generalized k-Fibonacci hyperbolic functions
    (Charles Babbage Res Ctr, 2010) Ipek, Ahmet
    The hyperbolic Fibonacci function, which is the being extension of Binet's formula for the Fibonacci number in continuous domain, transform the Fibonacci number theory into continuous theory because every identity for the hyperbolic Fibonacci function has its discrete analogy in the framework of the Fibonacci number. In this new paper, it is defined three important generalizations of the k-Fibonacci sine, cosine and quasi-sine hyperbolic functions and then many number of concepts and techniques that we learned in a standard setting for the k-Fibonacci sine, cosine and quasi-sine hyperbolic functions is carried over to the generalizations of these functions.
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    On the Solutions of Some Linear Complex Quaternionic Equations
    (Hindawi Ltd, 2014) Bolat, Cennet; Ipek, Ahmet
    Some complex quaternionic equations in the type AX - XB - C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.
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    On the solutions of the quaternion interval systems [x] = [A][x] plus [b]
    (Elsevier Science Inc, 2014) Bolat, Cennet; Ipek, Ahmet
    It is known that linear matrix equations have been one of the main topics in matrix theory and its applications. The primary work in the investigation of a matrix equation (system) is to give solvability conditions and general solutions to the equation(s). In the present paper, for the quaternion interval system of the equations defined by [x] = [A][x] + [b], where [A] is a quaternion interval matrix and [b] and [x] are quaternion interval vectors, we derive a necessary and sufficient criterion for the existence of solutions [x]. Thus, we reduce the existence of a solution of this system in quaternion interval arithmetic to the existence of a solution of a system in real interval arithmetic. (C) 2014 Elsevier Inc. All rights reserved.
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    On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries
    (Elsevier Science Inc, 2011) Ipek, Ahmet
    This paper is an improving of the work [S. Solak, On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comp. 160 (2005), 125-132], in which the lower and upper bounds for the spectral norms of the matrices A = [F-(mod(j-i,F-n))](i,j-1)(n) and B = [L-(mod(j-i,L-n))](i,j-1)(n) are established. In this new paper, we compute the spectral norms of these matrices. (C) 2010 Elsevier Inc. All rights reserved.
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    Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm
    (Pergamon-Elsevier Science Ltd, 2012) Ipek, Ahmet
    Let S = {x(1), . . . , x(n)} be a set of n distinct positive integers. The n x n matrix having the greatest common divisor (x(i), x(j)) of x(i) and x(j) as its i, j-entry is called the greatest common divisor (GCD) matrix defined on S, denoted by ((x(i), x(j))), or abbreviated as (S). The n x n matrix (S-1) = (g(ij)), where g(ij) = 1/(x(i),x(j)) is called the reciprocal greatest common divisor (GCD) matrix on S. In this paper, we present upper bounds for the spectral condition numbers of the reciprocal GCD matrix (S-1) and the GCD matrix (S) defined on S = {1, 2, . . . , n}, with n >= 2, as a function of Euler's phi function and n. (C) 2011 Elsevier Ltd. All rights reserved.