On the Generalized Order-k Jacobsthal and Jacobtshal-Lucas Numbers

Abstract

The classic Jacobsthal numbers were generalized to k sequences of the generalized order-k Jacobsthal numbers and then have been studied by several authors. In this paper, we explain that all of these studies used an incorrect version of order-k Jacobsthal numbers for reasons and give the correct definition of order-k Jacobsthal numbers. Further, we introduce the compatible generalized order-k Jacobsthal-Lucas numbers with the generalized order-k Jacobsthal numbers. Next, we give some properties of order-k Jacobsthal numbers and order-k Jacobsthal-Lucas numbers, including generating matrix, generalized Binet’s formula, and elementary matrix identities. Further, we investigate specific examples for our results and give special identities, i.e., sum formula and interrelationships between these sequences.

Keywords

• Generalized order-k sequence
• Jacobsthal sequence
• trace of matrix
• Binet
• Jacobsthal-Lucas sequence.

References

1. Aydın F.T., On generalizations of the Jacobsthal sequence, Notes on Number Theory and Discrete Mathematics, 24(1), 120-135, 2018.
2. Catarino P., Vasco P., Campos H., Aires A.P., Borges A., New families of Jacobsthal and Jacobsthal- Lucas numbers, Algebra and Discrete Mathematics, 20(1), 40-54, 2015.
3. Cook C.K., Bacon M.R., Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41, 27-39, 2013.
4. Da¸sdemir A., On the Jacobsthal numbers by matrix method, S¨uleyman Demirel University Faculty of Arts and Science Journal of Science, 7(1), 69-76, 2012.
5. Da¸sdemir A., A study on the Jacobsthal and Jacobsthal-Lucas numbers by matrix method, Dicle University Journal of the Institute of Natural and Applied Sciences, 3(1), 13-18, 2014.
6. Da¸sdemir A., Mersene, Jacobsthal, and Jacobsthal-Lucas numbers with negative subscripts, Acta Mathematica Universitatis Comenianae, 88(1), 142-156, 2019.
7. Falcon S., On the k -Jacobsthal numbers, American Review of Mathematics and Statistics, 2(1), 67-77, 2014.
8. Horadam A.F., A generalized Fibonacci sequence, American Mathematical Monthly, 68(5), 455-459, 1961.

9. Horadam A.F., Generating functions for powers of a certain generalised sequence of numbers, Duke Mathematical Journal, 32(3), 437-446, 1965.
10. Horadam A.F., Jacobsthal representation numbers, Fibonacci Quarterly, 34, 40-54, 1996.
11. Koken F., Bozkurt D., On the Jacobsthal numbers by matrix methods, International Journal of Contemporary Mathematical Sciences, 3(13), 605-614, 2008.
12. Koken F., Bozkurt D., On the Jacobsthal-Lucas numbers by matrix methods, International Journal of Contemporary Mathematical Sciences, 3(33), 1629-1633, 2008.
13. Koshy T., Elementary Number Theory with Applications, Harcourt Academic Press, 2002.
14. Machenry T., Wong K., Degree k linear recursions mod(p) and number fields, Rocky Mountain Journal of Mathematics, 41(4), 1303-1327, 2011.
15. Miles E.P., Generalized Fibonacci numbers and associated matrices, American Mathematical Monthly, 67(8), 745-752, 1960.
16. Stakhov A.P., Introduction into Algorithmic Measurement Theory, Moscow: Soviet Radio, 1977.
17. Stakhov A., Rozin B., On a new class of hyperbolic functions, Chaos, Solitons and Fractals, 23(2), 379-389, 2005.
18. Soykan Y., Ta¸sdemir E., Okumu¸s ˙I., On dual hyperbolic numbers with generalized Jacobsthal numbers components, Indian Journal of Pure and Applied Mathematics, 54(3), 824-840, 2023.
19. Uygun S., Owusu E., A new generalization of Jacobsthal numbers (bi-periodic Jacobsthal sequences), Journal of Mathematical Analysis, 7(5), 28-39, 2016.
20. Vajda S., Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications, Courier Corporation, 2008.
21. Yilmaz F., Bozkurt D., The generalized order-k Jacobsthal numbers, International Journal of Contemporary Mathematical Sciences, 4(34), 1685-1694, 2009.