We prove a Nullstellensatz for the ring of polynomial functions in n non-commuting variables over Hamilton's real quaternions. also characterize generalized identities which hold quaternions, and more generally, any division algebra .

In this study, generalized k-order Fibonacci hybrid quaternion is defined. We give recurrence relation, generating function, the summation formula and some properties for these quaternions. Furthermore, matrix representation quaternions determined. The Q_k defined numbers given By means of another matrices, several identities are also obtained.

In this present study, we intend to determine the Padovan, Perrin and Pell-Padovan dual quaternions with nonnegative negative subscripts. line purpose, construct some new properties such as; special determinant equalities, recurrence relations, matrix formulas, Binet-like generating functions, exponential summation binomial for these quaternions.

In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions present some of their properties. Some identities are derived for these polynomials. Furthermore, give the Binet formulas, Catalan, Cassini, d’Ocagne identity generating exponential function quaternions.