Algebraic properties of some special quaternion sequences and polynomials

In this study, firstly, historical process of some special number sequences, polynomials and quaternions are examined. Then, the Q-matrix that forms the basis of the generating matrices is reminded. In the second part, basic information about special number sequences, polynomials and quaternions are given. In the third part, bivariate Fibonacci quaternion polynomials and bivariate Lucas quaternion polynomials are introduced and their basic properties are studied. Moreover, Binet formulas, generating functions, special identities such as Catalan, Cassini, d'Ocagne and various sum formulas and matrix representations of these quaternion polynomials are given with the proofs. In the next part, motivating the Binet formula of Horadam quaternions which is a generalization of quaternion sequences, various sum and binomial formulas, identities and matrix representations are obtained. In the last part, results and suggestions are presented.