On generalized Jacobsthal and Jacobsthal-Lucas numbers
Abstract
Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.
Keywords
Jacobsthal numbers; Jacobsthal–Lucas numbers; generalized Jacobsthal numbers; Binet formula
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Zakład Matematyki Dyskretnej, Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
Zakład Matematyki Dyskretnej, Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
https://orcid.org/0000-0002-8776-5270
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