New families of Jacobsthal and Jacobsthal-Lucas numbers

In this paper we present new families of sequences that generalize the Jacobsthal and the Jacobsthal-Lucas numbers and establish some identities. We also give a generating function for a particular case of the sequences presented. Introduction Several sequences of positive integers were and still are object of study for many researchers. Examples of these sequences are the well known Fibonacci sequence and the Lucas sequence, both related with the golden mean, with so many applications in diverse fields such as mathematics, engineering, biology, physics, architecture, stock market investing, among others (see [9] and [17]). About these and other sequences like Pell sequence, Pell-Lucas sequence, Modified Pell sequence, Jacobsthal sequence and the Jacobsthal-Lucas sequence, among others, there is a vast literature where several properties are studied and well known identities are derived, see for example, [13, 18–20]. In 1965, Horadam studied some properties of sequences of the type, wn(a, b; p, q), where a, b are nonnegative integers and p, q are arbitrary 2010 MSC: 11B37, 11B83, 05A15.