On Generalized Jacobsthal and Jacobsthal-Lucas Numbers

• Autorzy: Dorota Bród , Adrian Michalski
• Języki publikacji: EN

Abstrakty

EN

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.(original abstract)

Słowa kluczowe

EN

Mathematics; Number theory

PL

Matematyka; Teoria liczb

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2022
• Tom: 36 (2)
• Strony: 115--128

Twórcy

autor

Dorota Bród

• Rzeszow University of Technology,, Rzeszów, Poland

autor

Adrian Michalski

• Rzeszow University of Technology,, Rzeszów, Poland

Bibliografia

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• Halici S. and Uysal M., A study on some identities involving (s_k,t)-Jacobsthal numbers, Notes Number Theory Discrete Math. 26 (2020), no. 4, 74-79.
• Horadam A.F., Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), no. 1, 40-54.
• Jhala D., Sisodiya K., and Rathore G.P.S., On some identities for k-Jacobsthal numbers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 12, 551-556.
• Köken F. and Bozkurt D., On the Jacobsthal-Lucas numbers by matrix method, Int. J. Contemp. Math. Sci. 3 (2008), no. 33, 1629-1633.
• Sloane N.J.A., The On-Line Encyclopedia of Integer Sequences. Avaliable at https://oeis.org/book.html.
• Szynal-Liana A., Włoch A., and Włoch I., On generalized Pell numbers generated by Fibonacci and Lucas numbers, Ars Combin. 115 (2014), 411-423.
• Wani A.A., Catarino P., and Halici S., On a study of (s,t)-generalized Pell sequence and its matrix sequence, Punjab Univ. J. Math. (Lahore) 51 (2019), no. 9, 17-32.

Identyfikatory

DOI

10.2478/amsil-2022-0011