3-Parameter Generalized Quaternions and Dual Quaternions with Repunit Number Components

In this paper, our first aim is to combine the 3-parameter generalized quaternions, which are a general form of the quaternion algebra according to 3-parameters, and repunit numbers, which are created by integer numbers that are expressed in the decimal system as the repetition of the unit. That is, we determine a special new number system called 3-parameter generalized quaternions with repunit number components. Then, we construct a Maple code of this special number sequence. Additionally, we get some new and classical well-known equations such as Binet formulas, generating functions (usual, exponential, and Poisson), polar representation, and Cassini and Catalan identities. In addition to these, we express determinant property, characteristic polynomial, characteristic equation, eigenvalues, and eigenvectors concerning the matrix representation of repunit 3-parameter generalized quaternions. Next, the other purpose is to combine the dual quaternions with the repunit numbers. With the same manner of first aim and structure, we construct a new recurrence sequence, which is called dual quaternions with repunit number components.