A new member of the Pell sequences: The pseudo-Pell sequence
DOI:
https://doi.org/10.24193/subbmath.2025.2.01Keywords:
Pell and Pell-Lucas numbers, pseudo-Pell and Pell-Lucas numbersAbstract
In this study, we define a new family of the Pell numbers and establish some properties of the relation to the ordinary Pell numbers. We give some identities the pseudo-Pell numbers. Moreover, we obtain the Binet’s formula, generating function formula and some formulas for this new type numbers. Moreover, we give the matrix representation of the pseudo-Pell numbers.
Mathematics Subject Classification (2010): 11B37, 11B83, 11C20.
Received 20 April 2024; Accepted 10 January 2025.
References
1. Bacani, J.B., Rabago, J.F.T., On generalized Fibonacci numbers, Appl. Math. Sci., (2015).
2. Bravo, J.J., Herrera, J.L., Luca, F., On a generalization of the Pell sequence, Math. Bohem., 146(2021), no. 2, 199-213.
3. Catarino, P., On some identities and generating functions for k-Pell numbers, Int. J. Math. Anal., 7(2013), no. 38, 1877-1884.
4. Cigler, J., A new class of q-Fibonacci polynomials, Electron. J. Combin., 10(2003), no. 1, 1-15.
5. Cigler, J., Some beautiful q-analogs of Fibonacci and Lucas polynomials, arXiv:1104.2699v1.
6. Edson, M., Yayenie, O., A new generalization of Fibonacci sequence and extended Binet's formula, Integers Electron. J. Comb. Number Theor., 9(2009), no. 6, 639-654.
7. Falcon, S., Plaza, A., The k-Fibonacci sequence and the Pascal 2-triangle, Chaos Solitons Fractals, 33(2007), no. 1, 38-49.
8. Ferns, H.H., Pseudo-Fibonacci numbers, Fibonacci Quart., 6(1968), no. 6, 305-317.
9. Gould, H.W., A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart., 19(1981), no. 3, 250-257.
10. Ipek, A., Ari, K., Türkmen, R., The generalized (s; t)-Fibonacci and Fibonacci matrix sequences, Transylvanian Journal of Mathematics and Mechanics, 7(2015), no. 2, 137-148.
11. Kalman, D., Generalized Fibonacci numbers by matrix methods, Fibonacci Quart., 20(1982), no. 1, 73-76.
12. Kalman, E., The generalized Pell (p; i)-numbers and their Binet formulas, combinatorial representations, sums, Chaos Solitons Fractals, 40(2009), no. 4, 2047-2063.
13. Koçer, E.G., Tuğlu, N., The Binet formulas for the Pell and Pell-Lucas p-numbers, Ars Combin., 85(2007), 3-17.
14. Koshy, T., Fibonacci and Lucas numbers with applications I, John Wiley and Sons, 2001.
15. Koshy, T., Pell and Pell-Lucas numbers with applications, New York, Springer, 2014.
16. Koshy, T., Fibonacci and Lucas numbers with applications II, John Wiley and Sons, 2019.
17. Kuloğlu, B. Özkan, E., Shannon, A.G., p-analogue of biperiodic Pell and Pell-Lucas polynomials, Notes on Number Theory and Discrete Mathematics, 29(2023), no. 2, 336-347.
18. Lee, G., Aşçi, M., Some properties of the (p; q)-Fibonacci and (p; q)-Lucas polynomials, J. Appl. Math., 1(2012), 1-18.
19. Lee, G.Y., Lee, S.G., A note on generalized Fibonacci numbers, Fibonacci Quart., 33(1995), no. 3, 273-278.
20. Makate, N., Rattanajak, P., Mongkhol, B., Bi-periodic k-Pell sequence, International Journal of Mathematics and Computer Science, 19(2024), no. 1, 103-109.
21. Motta, W., Rachidi, M., Saeki, O., On ∞-generalized Fibonacci sequences, Fibonacci Quart., 37(1999), 223-232.
22. Nalli, A., Haukkanen, P., On generalized Fibonacci and Lucas polynomials, Chaos Solitons Fractals, 42(2009), 3179-3186.
23. Prodinger, H., Summing a family of generalized Pell numbers, Ann. Math. Sil., 35(2021), no. 1, 105-112.
24. Saba, N., Boussayoud, A., Ordinary generating functions of binary products of (p; q)-modified Pell numbers and k-numbers at positive and negative indices, Journal of Science and Arts, 3(2020), no. 52, 627-648.
25. Spreafico, E.V.P., Rachidi, M., On generalized Pell numbers of order r ≥ 2, Trends Comput. Appl. Math., 22(2021), 125-138.
26. Srisawat, S., Sriprad, W., On the (s; t)-Pell and (s; t)-Pell-Lucas numbers by matrix methods, Ann. Math. Inform., 46(2016), 195-204.
27. Stakhov, A., Rozin, B., Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos Solitons Fractals, 27(2006), no. 5, 1162-1177.
28. Uygun, Ş, Karataş, H., A new generalization of Pell-Lucas numbers, Communications in Mathematics and Applications, 10(2019), no. 3, 469-479.
29. Waddill, M.E., Sacks, L., Another generalized Fibonacci sequence, Fibonacci Quart., 5(1967), no. 3, 209-222.
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