On the 3-parameter generalized quaternions with generalized tribonacci numbers components

Abstract

In this paper, we aim to combine 3-parameter generalized quaternions (shortly 3PGQs), which are a general form of the quaternion algebra according to 3-parameters, and generalized Tribonacci number (shortly GTNs), which are also quite a big special number family for third-order recurrence sequences and most general form of all of the third-order recurrence sequences. Namely, we investigate a special new number system called 3-parameter generalized quaternions with generalized Tribonacci numbers components (shortly 3PGQs with GTN components) with both nonnegative and negative subscripts and examine some special cases of them. Then, we construct a Maple code of this special number family. Moreover, we obtain some new and classical well-known equations such as; Binet formulas, generating function, exponential generating function, Poisson generating function, summation formulas, polar representation, and matrix equation. In addition to these, we give also determinant, characteristic polynomial, characteristic equation, eigenvalues, and eigenvectors concerning the matrix representation of 3PGQs with GTN components.