Abstarct: Over the past four decades, quantum algebra has been the focus of numerous studies in various fields of physics and mathematics. In this thesis, we examine the deformed formalism. As is well known, the deformed formalism always represents a more general case of its corresponding original theory, such that the original formalism can be recovered by allowing the deformation parameter to approach a specific value. In our study of deformed formalisms, we introduce a new algebra. This thesis investigates how this particular algebra arises from quantum mechanics. We derive the Schrödinger and relativistic equations, solve them for a class of specific interactions, and extend the formalism to N particles. Using a fermionic system, we compute the spectrum and eigenvalues for these particles. Initially, we introduce the kappa-deformed formalism, and then we study trigonometric functions, logarithmic functions, weight functions, and other properties within the context of the kappa-deformed formalism. With the presence of the kappa parameter in this formalism, we can apply it to theoretical models with certain undefined parameters and, ultimately, determine these parameters. Additionally, employing the kappa-deformed formalism, we calculate the thermodynamic properties of the harmonic oscillator problem. We demonstrate that the kappa parameter can be a tuning factor for thermodynamic properties. In the case of white dwarfs, where degeneracy pressure balances gravitational pressure, we calculate and examine the radius of white dwarfs at various densities, achieving good agreement with other results. Furthermore, the Lane-Emden equation, a fundamental equation in mathematical physics, describes the density distribution within a polytropic star in hydrostatic equilibrium. Using the standard Lane-Emden equation and its modified version with gravitational deviation, we compute the radius of white dwarfs. At each step, we compare the results in the limiting case, where the kappa parameter approaches zero, with those of the original theories.