On the k-Vieta-Pell and k-Vieta-Pell-Lucas Sequences

Year 2025, Volume: 10 Issue: 2, 113 - 128, 30.08.2025

Hakan Akkuş , Engin Özkan

https://doi.org/10.30931/jetas.1562212

Cited By: 1

https://izlik.org/JA42FH58CD

Abstract

The aim of this paper is to define the k-Vieta-Pell and k-Vieta-Pell-Lucas sequences, and some terms of these sequences are given. Then, we find the relations between the terms of the k-Vieta-Pell and k-Vieta-Pell-Lucas sequences. Also, we give the summation formulas, generating functions, etc. We also derive the Binet formulas using two different approaches. The first is in the known classical way and the second is with the help of the sequence's generating functions. Moreover, we calculate the special identities of these sequences like Catalan and Melham. Finally, we examine the relations between the k-Vieta-Pell sequence and various other sequences, including Fibonacci, Pell, and Chebyshev polynomials of the first kind. Similarly, we analyze the k-Vieta-Pell-Lucas sequence in relation to Lucas, Pell-Lucas numbers, Chebyshev polynomials of the second kind, and other sequences. In addition, for special k values, these sequences are associated with the sequences in OEIS.

Keywords

Vieta polynomials , Generating function , Pell number , Cassini Identity , Binet formula

References

• [1] Aydınyüz, S., Aşcı, M., ‘‘The Moore-Penrose Inverse of the Rectangular Fibonacci Matrix and Applications to the Cryptology’’, Advances and Applications in Discrete Mathematics 40(2) (2023) : 195-211.
• [2] Turner, H.A., Humpage, M., Kerp, H., Hetherington, A.J., ‘‘Leaves and sporangia developed in rare non-Fibonacci spirals in early leafy plants’’, Science 380(6650) (2023) : 1188-1192.
• [3] Avazzadeh, Z., Hassani, H., Agarwal, P., Mehrabi, S., Javad Ebadi, M., Hosseini Asl, M. K., ‘‘Optimal study on fractional fascioliasis disease model based on generalized Fibonacci polynomials’’, Mathematical Methods in the Applied Sciences 46(8) (2023) : 9332-9350.
• [4] Otto, H.H., ‘‘Fibonacci Stoichiometry and Superb Performance of Nb16W5O55 and Related Super-Battery Materials’’, Journal of Applied Mathematics and Physics 10(6) (2022) : 1936-1950.
• [5] de Oliveira, R.R., Alves, F.R.V., ‘’An investigation of the bivariate complex fibonacci polynomials supported in didactic engineering: an application of Theory of Didactics Situations (TSD)’’, Acta Scientiae 21(3) (2019) : 170-195.
• [6] Akkuş, H., Üregen, R.N., Özkan, E., ‘‘A New Approach to 𝑘−Jacobsthal Lucas Sequences’’, Sakarya University Journal of Science 25(4) (2021) : 969-973.
• [7] Tastan, M., Özkan, E., ‘‘Catalan transform of the k-Pell, k-Pell-Lucas and modified k-Pell sequence’’, Notes On Number Theory and Discrete Mathematics 27(1) (2021) : 198-207.
• [8] Özkan, E., Akkus, H., ‘‘On k-Chebsyhev Sequence’’, Wseas Transactions On Mathematics 22 (2023) : 503-507.
• [9] Akkuş, H., Deveci, Ö., Özkan, E., Shannon, A.G., ‘‘Discatenated and lacunary recurrences’’, Notes on Number Theory and Discrete Mathematics 30(1) (2024) : 8-19.
• [10] Özkan, E., Akkuş, H., ‘‘Copper ratio obtained by generalizing the Fibonacci sequence’’, AIP Advances 14(7) (2024) : 1-11.
• [11] Horadam, A.F., ‘‘Vieta polynomials’’, Fibonacci Quarterly 40(3) (2002) : 223-232.
• [12] Shannon, A.G., Horadam, A.F. ‘‘Some relationships among Vieta, Morgan- Voyce and Jacobsthal polynomials’’, Applications of Fibonacci Numbers 8 (1999) : 307-323.
• [13] Robbins, N., ‘‘Vieta's triangular array and a related family of polynomials’’, International Journal of Mathematics and Mathematical Sciences 14 (1991) : 239-244.
• [14] Mason, J.C., Handscomb, D.C., ‘‘Chebyshev polynomials’’, CRC press, New York 2002.
• [15] Griggs, J.R., Hanlon, P., Odlyzko, A.M., Waterman, M.S., ‘‘On the number of alignments of k sequences’’, Graphs and Combinatorics 6(2) (1990) : 133-146.
• [16] Falcon, S., Plaza, A., ‘‘On the Fibonacci k-numbers’’, Chaos Solitons Fractals 32(5) (2007) : 1615–1624.
• [17] Falcon, S., ‘‘On the k-Lucas numbers’’, International Journal of Contemporary Mathematical Sciences 6(21) (2011) : 1039-1050.
• [18] Falcon, S., ‘‘Catalan Transform of the k-Fibonacci sequence’’, Communications of the Korean Mathematical Society 28(4) (2013) : 827-832.
• [19] Shannon, A.G., Akkuş, H., Aküzüm, Y., Deveci, Ö., Özkan, E., ‘‘A partial recurrence Fibonacci link’’, Notes on Number Theory And Discrete Mathematics 30(3) (2024) : 530-537.
• [20] Soykan, Y., Taşdemir, E., Okumuş, I., ‘‘A study on binomial transform of the generalized padovan Sequence’’, Journal of Science and Arts 22(1) (2022) : 63-90.
• [21] Özkan, E., Akkuş, H., ‘‘A New Approach to k-Oresme and k-Oresme-Lucas Sequences’’, Symmetry 16 (2024) : 1-12.
• [22] Koshy, T., ‘‘Fibonacci and Lucas Numbers with Applications’’, Volume 2. John Wiley & Sons (2019).
• [23] Özkan, E., Akkuş, H., ‘‘Generalized Bronze Leonardo sequence’’, Notes on Number Theory and Discrete Mathematics 30(4) (2024) : 811-824.
• [24] Akkuş, H., Özkan, E., ‘‘Generalization of the k-Leonardo Sequence and their Hyperbolic Quaternions’’, Mathematica Montisnigri 60 (2024) : 14-31.