In this paper, we introduce the hyperbolic k−Jacobsthal and k−Jacobsthal-Lucas quaternions. We present generating functions, Binet formula, Catalan's identity, Vajda's identity etc. for the hyperbolic k-Jacobsthal and k−Jacobsthal-Lucas quaternions.

We introduce Mersenne-Lucas hybrid numbers. We give the Binet formula, the generating function, the sum, the character, the norm and the vector representation of these numbers. We find some relations among Mersenne-Lucas hybrid numbers, Jacopsthal hybrid numbers, Jacopsthal-Lucas hybrid numbers and Mersenne hybrid numbers. Then we present some important identities such as Cassini identities for Mersenne-Lucas hybrid numbers

Highlights• This paper focuses on higher order Jacobsthal quaternions.• We examined the basic properties of these numbers in this study.• We have given some identities of these numbers.

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas–Eisenstein Series. First, we show that the Pell-Lucas–Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.

In this work, we investigate the hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions. We give Binet’s Formula, Cassini’s identity, Catalan’s identity, d’Ocagne identity, generating functions of the hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions. Also, we present many properties of these octonions.