The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.

It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the $4\times 4$ matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

In this study, we consider Kn := circ (GN1,GN2,...,GNn) circulant matrices whose entries are the Gaussian Nickel Fibonacci numbers GN1,GN2,...,GNn. Then, we compute determinants of Kn by exploiting Chebyshev polynomials of the second kind. Moreover, we obtain Cassini’s identity and the D’Ocagne identity for the Gaussian Nickel Fibonacci numbers.

This paper aims to present a method for constructing the second order Pell and Pell-Lucas numbers and the third order Pell and Pell-Lucas numbers. Moreover, we obtain the De Moivre-type identities for these numbers. In addition, we define a Pell sequence with new initial conditions and give some identities for these third order Pell numbers such as Binet's formulas, generating functions, sums.

The aim of this study is to investigate the oil prices, which have crucial impact of an economy, using new ratios called Nickel ratios instead of the golden ratios on technical analysis. The Nickel ratios are developed considering Nickel Fibonacci sequence. This study is the first to use Nickel ratios in technical analysis in economics and finance. In this study, graphs comprising of weekly, daily, $4-$hour and $30-$minute periods are analyzed using Nickel ratios in Fibonacci retracement, fan, arcs and time zones applications, and the results are compared with the golden ratio obtained from the Fibonacci number sequence. In addition, the support and resistance points obtained from Nickel ratios have more significant levels than the golden ratio. The retracement, fan, arcs and time zones graphs with weekly, daily, four hourly and half-hourly data based on the golden and Nickel ratios show that the levels regarding the Nickel ratios confirm more significant points in comparison with the levels regarding the golden ratios. Finally, more efficient results are observed when the ratios of golden and Nickel are considered together.