A Note on Modified $k$-Pell Hybrid Numbers

Abstract

Modified $k$-Pell hybrid sequence is defined in this paper. The Binet formula and the generating functions for modified k-Pell hybrid numbers are also presented. By using the Binet formula, some properties involving this sequence, including Catalan's, Cassini's and d'Ocagne's identities, are obtained.                                                                                                                                                                                                                                                                                                                                                                                          

Keywords

• Modified $k$-Pell numbers
• Hybrid numbers
• Recurrence relations
• Generating functions
• Binet's formula

Supporting Institution

Portuguese Funds through FCT

Project Number

UIDB/00013/2020

References

1. [1] N. Bicknell, A primer on the Pell sequence and related sequence, Fibonacci Quart., Vol:13, No.4 (1975), 345-349.
2. [2] P. Catarino, On some identities and generating functions for k-Pell numbers, Int. J. Math. Anal. (Ruse), Vol:7, No.38 (2013), 1877-1884.
3. [3] P. Catarino and P. Vasco, On Some Identities and Generating Functions for k-Pell-Lucas Sequence, Appl. Math. Sci., Vol:7, No.98 (2013), 4867-4873.
4. [4] P. Catarino and P. Vasco, Modified k-Pell Sequence: Some Identities and Ordinary Generating Function, Appl. Math. Sci., Vol:7, No.121 (2013), 6031-6037.
5. [5] P. Catarino, A note involving two-by-two matrices of the k-Pell and k-Pell-Lucas sequences, Int. Math. Forum, Vol:8, No.32 (2013), 1561-1568.
6. [6] P. Catarino, On k-Pell hybrid numbers, Journal of Discrete Mathematical Sciences and Cryptography, Vol:22, No.1 (2019), 83-89.
7. [7] A. Dasdemir, On the Pell, Pell-Lucas and Modified Pell Numbers by Matrix Method, Appl. Math. Sci. (Ruse), Vol:5, No.64 (2011), 3173-3181.
8. [8] J. Ercolano, Matrix generators of Pell sequences, Fibonacci Quart., Vol:17, No.1 (1979), 71-77.

9. [9] S. Halici and S. Çürük, On Dual k-Pell Bicomplex Numbers and some Identities including them, Fundamental Journal of Mathematics and Applications, Vol:3, No.1 (2020), 86-93.
10. [10] S. Halici, Some sums formulae for products of terms of Pell, Pell-Lucas and Modified Pell sequences, SA¨U Fen Billimleri Dergisi, Vol:15, Cilt, 2, Say1, (2011), 151-155.
11. [11] A. F. Horadam, Applications of Modified Pell Numbers to Representations, Ulam Quarterly, Vol:3, No.1 (1994), 34-53.
12. [12] A. F. Horadam, Pell identities, Fibonacci Quart., Vol:9, No.3 (1971), 245-252, 263.
13. [13] C. Kızılates¸, A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos Solitons Fractals, Vol:130 (2020), https://doi.org/10.1016/j.chaos.2019.109449
14. [14] A. Özkoc, A. Tekcan and G. K. Go¨zeri, Triangular and Square Triangular Numbers involving Generalized Pell Numbers, Utilitas Mathematica, Vol:102 (2017), 231-254.
15. [15] M. Özdemir, Introduction to Hybrid Numbers, Adv. Appl. Clifford Algebras Vol:28, No.11 (2018) https://doi.org/10.1007/s00006-018-0833-3.
16. [16] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.
17. [17] A. Szynal-Liana and I. Włoch, On Jacobsthal and Jacobsthal-Lucas Hybrid Numbers, Annales Mathematicae Silesianae, Vol:33, No.1 (2019), 276-283.
18. [18] A. Szynal-Liana and I. Włoch, On Pell and Pell-Lucas Hybrid Numbers, Commentat Math, Vol:58 (2018), 11-17.
19. [19] A. Szynal-Liana, The Horadam Hybrid Numbers, Discuss Math Gen Algebra Appl, Vol:38, No.1 (2018), 91-98.
20. [20] P. Vasco and P. Catarino, Sums and Products involving terms of k-Pell, k-Pell-Lucas and Modified k-Pell sequences, JP J. Algebra Number Theory Appl., Vol:32, No.2 (2014), 87-98.
21. [21] P. Vasco, P. Catarino, H. Campos, A. P. Aires and A. Borges, k-Pell, k-Pell-Lucas and Modified k-Pell numbers: Some identities and Norms of Hankel Matrices, Int. J. Math. Anal. (Ruse) Vol:9, No.1 (2015), 31-37.