Let us define A = C r (a 0 , a 1 , . . . , a n-1 ) to be a n × n r-circulant matrix. The entries in the first row of A = C r (a 0 , a 1 , . . . , a n-1 ) are(i = 0, 1, 2, . . . , n -1), where P i and Q i are the ith Pell and Pell-Lucas numbers, respectively. We find some bounds estimation of the spectral norm for r-Circulant matrices with Pell and Pell-Lucas numbers.

Abstract

In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-hybrid Pell and Pell-Lucas numbers. We also give negaGaussian-hybrid Pell and Pell-Lucas numbers, the characteristic number and the type number of Gaussian-hybrid Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function formula, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for the Gaussian-hybrid Pell and Pell-Lucas numbers. Some algebraic proporties of Gaussian-hybrid Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix representation of Gaussian-hybrid Pell and Pell-Lucas numbers.

In this study, we define a new type of Pell and Pell-Lucas numbers which are called biGaussian Pell and biGaussian Pell-Lucas numbers. We also give the relationship between negabiGaussian Pell and Pell-Lucas numbers and bicomplex Pell and Pell-Lucas numbers. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sums formulas for these new type numbers. Some algebraic proporties of biGaussian Pell and Pell-Lucas numbers which are connected between biGaussian numbers and Pell and Pell-Lucas numbers are investigated. Moreover, we give the matrix representation of biGaussian Pell and Pell-Lucas numbers.

Let us define () r ij A H a = = to be n n × r-Hankel matrix. The entries of matrix A are 2 n i j F F + − = or 2 n i j L F + − = where n F and n L denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact value of matrix A's spectral norm. These kinds of matrices have connections with signal and image processing, time series analysis and many other problems.