In the present work, we define harmonic complex balancing numbers by considering well-known balancing numbers and inspiring harmonic numbers. Mainly, we investigate some of their basic characteristic properties such as the Binet formula and Cassini identity, etc. In addition, one type of symmetric matrix family whose entries are harmonic complex balancing numbers is constructed. Additionally, some linear algebraic properties are obtained. Furthermore, some inequalities are stated by exploiting the well-known inequalities between various matrix norms. Finally, we illustrate the results with some numerical examples.

At this paper, we describe Gaussian Oresme numbers taking into account the Oresme numbers. Furthermore, we investigate their some basic characteristic properties such as Binet formula and Cassini identity, etc. Moreover, we define quaternions with Gaussian Oresme coefficients and obtain their some spectacular properties.

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.

In the present study, inspired by the studies in the literature, we consider Min matrix and its Hadamard exponential matrix family whose elements are Chebyshev polynomials of the first kind. Afterwards, we examine their various linear algebraic properties and obtain some inequalities. Furthermore, we shed light on the results we obtained to boost the clarity of our paper with the illustrative examples. In addition to all these, we give two MATLAB-R2023a codes that compute the Min matrix and the Hadamard exponential matrix with Chebyshev polynomials of the first kind entries, as well as calculate some matrix norms.