Bi-periodic jacobsthal and jacobsthal lucas integer and matrix sequences

In this research, we bring into light four strong generalizations on Jacobsthal and Jacobsthal Lucas sequences which we shall call the bi-periodic Jacobsthal and Jacobsthal Lucas sequences. Some relations between these sequences will be examined. Their matrix representations shall constitute the last two generalizations. We proceed to find the generating functions as well as the Binet formulas for the bi-periodic Jacobsthal and Jacobsthal Lucas sequences. The well-known Cassini, Catalan and the D'ocagne identities as well as some related binomial summation formulas are also given. In addition, we established a good number of relationships between our generalized bi-periodic Jacobsthal and Jacobsthal Lucas sequences. In the third generalization, we bring into light the matrix representation of bi-periodic Jacobsthal sequence. We then proceed to obtain the nth general term of this new matrix sequence. Cassini or Simpson's formula, the generating function as well as the Binet formula are also given. Some new properties together with some summation formulas for this new generalized matrix sequence are also given. Lastly, we bring into light the matrix representation of bi-periodic Jacobsthal Lucas sequence. Simpson's formula, the generating function as well as the Binet formula with some new properties for this generalized matrix sequence are given.