On the Spectrum and Spectral Norms of -Circulant Matrices with Generalized -Horadam Numbers Entries

This work is concerned with the spectrum and spectral norms of -circulant matrices with generalized -Horadam numbers entries. By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficient condition for an -circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal -circulant matrix with generalized -Horadam numbers, which generalize and improve the known results. 1. Introduction There is no doubt that the -circulant matrices have been one of the most interesting research areas in computation mathematics. It is well known that these matrices have a wide range of applications in signal processing, digital image disposal, coding theory, linear forecast, and design of self-regress. There are many works concerning estimates for spectral norms of -circulant matrices with special entries. For example, Solak [1] established lower and upper bounds for the spectral norms of circulant matrices with Fibonacci and Lucas numbers entries. subsequently, Ipek [2] investigated some improved estimations for spectral norms of these matrices. Bani-Domi and Kittaneh [3] established two general norm equalities for circulant and skew circulant operator matrices. Shen and Cen [4] gave the bounds of the spectral norms of -circulant matrices whose entries are Fibonacci and Lucas numbers. In [5] they defined -circulant matrices involving -Lucas and -Fibonacci numbers and also investigated the upper and lower bounds for the spectral norms of these matrices. Recently, Yazlik and Taskara [6] define a generalization of the special second-order sequences such as Fibonacci, Lucas, -Fibonacci, -Lucas, generalized -Fibonacci and -Lucas, Horadam, Pell, Jacobsthal, and Jacobsthal-Lucas sequences. For any integer number , the generalized -Horadam sequence is defined by the following recursive relation: where and are scaler-value polynomials, . The following are some particular cases.(i)If , and , , the -Fibonacci sequence is obtained: (ii)If , and , , the -Lucas sequence is obtained: (iii)If , and , , the Fibonacci sequence is obtained: (iv)If , and , , the Lucas sequence is obtained: (v)If , and , , the Jacobsthal sequence is obtained: In [7], the authors present new upper and lower bounds for the spectral norm of an -circulant matrix , and they study the spectral norm of circulant matrix with generalized -Horadam numbers in [8]. In this paper, we first give an explicit formula for the eigenvalues of -circulant matrix with generalized -Horadam numbers