Estimation Some Inequalities for the Spectral Norm of Various Specal Matrices with generalized Pell numbers

In this article, we first define bi-periodic Pell sequence which is a generalization of Pell numbers consisting of two positive real numbers. We present some various sum formulas for the bi-periodic Pell numbers. After giving the definitions of r-circulant, symmetric rcirculant and geometric circulant matrices, we construct these circulant matrices with the generalized Pell sequence. Then, we investigate some inequalities for the bounds of the spectral norms of r-circulant, symmetric r-circulant and geometric circulant matrices with the generalized Pell numbers by three different ways. Finally, we give several corollaries related to norms of Hadamard and Kronecker products of these circulant matrices. The eigenvalues are also established for the r-circulant matrices whose entries are the generalized Pell numbers.