This study provides a broad overview of the generalization of the various quaternions, especially in the context of its enhancing importance in the disciplines of mathematics and physics. By the help of bicomplex numbers, in this paper, we define the bicomplex generalized k Horadam quaternions. Fundamental properties and mathematical preliminaries of these quaternions are outlined. Finally, we give some basic conjucation identities, generating function, the Binet formula, summation formula, matrix representation and a generalized identity, which is generalization of the well-known identities such as Catalan's identity, Cassini's identity and d'Ocagne's identity, of the bicomplex generalized k Horadam quaternions in detail.

The purpose of this paper is to provide a broad overview of the generalization of the various dual complex number sequences, especially in the disciplines of mathematics and physics. By the help of dual numbers and dual complex numbers, in this paper, we de…ne the dual complex generalized k Horadam numbers. Furthermore, we investigate the Binet formula, generating function, some conjugation identities, summation formula and a theorem which is generalization of the Catalan's identity, Cassini's identity and d'Ocagne's identity.

In this paper, we investigate the effect of Magnus Expansion Method which is based on Lie groups and Lie algebras on some specific nonlinear differential equations which are Liénard and Isothermal Gas Spheres Equation. Moreover, we obtain the numerical results and then we present approximate, exact values and absolute error norms in detail.

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In this paper, we introduce the Wilker$-$Anglesio's inequality and parameterized Wilker inequality for the $k-$Fibonacci hyperbolic functions using classical analytical techniques.

In this paper, Magnus expansion method which is based on Lie groups and Lie algebras is presented to solve singularly perturbed boundary value problems having boundary layers. Numerical results are obtained by using different step sizes and small ε values which makes the problem stiff or nonstiff. In addition, maximum errors norms with L ∞ are tabulated in detail to show efficiency of the method.(2) Under these assumptions (2), problem (1) has a unique solution having boundary layers.

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This study introduces the modified generalized Fibonacci and Lucas 2k−ions which are the generalizations of several quaternions, octonions and higher order dimensional algebras. We give the generating functions, the Binet formulas and well-known identities such as Catalan’s identity and Cassini’s identity for the modified generalized Fibonacci and Lucas 2k−ions.