Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences

José Chimpanzo; Paula Maria Machado Cruz Catarino; María Victoria Otero Espinar

doi:10.32323/ujma.1393569

EN

Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences

Abstract

This article studies the bidimensional versions of four number sequences: balancing, Lucas-balancing, Lucas-cobalancing and cobalancing. Some of the identities and generating functions were presented.

Keywords

Bidimensional balancing, Bidimensional cobalancing, Bidimensional Lucas-balancing, Bidimensional Lucas-cobalancing, Generating functions, Some identities

References

[1] G. K. Gözeri, On Pell, Pell-Lucas, and balancing numbers, J. Inequal. Appl., 3 (2018). https://doi.org/10.1186/s13660-017-1599-1
[2] B. K. Patel, N. Irmak, P. K. Ray, A. Zaharescu, Incomplete balancing and Lucas-balancing numbers, Math. Rep. (Bucur.), 20(1) (2018), 59-72.
[3] G. K. Panda, T. Komatsu, R. K. Davala, Reciprocal sums of sequences involving balancing and Lucas-balancing numbers, Math. Rep. (Bucur.), 20(2) (2018), 201-214.
[4] C. Levesque, On m-th order linear recurrences, Fibonacci Quart., 23(4) (1985), 290–293.
[5] M. Vieira, V. Alves, P. Catarino, Relacoes bidimensionais e identidades da sequˆencia de Leonardo, ReviSeM, 2 (2019), 156–173. https://doi.org/10.34179/revisem.v4i2.11863
[6] C. Bezerra, V. Alves, M. Vieira, Relacoes recorrentes bidimensionais e tridimensionais de Narayana, C.Q.D-Revista Eletronica Paulista de Matem´atica, 18 (2020), 12–28. https://doi.org/10.21167/cqdvol18ic202023169664fhcbfrva1228
[7] R. R. de Oliveira, V. Alves, B. Paiva, Identidades bi e tridimensionais para os n´umeros de Fibonacci na forma complexa, C.Q.D-Revista Eletronica Paulista de Matem´atica, 11 (2017), 91–106. https://doi.org/10.21167/cqdvol11ic201723169664rrofrvarebp91106
[8] R. R. de Oliveira, V. Alves, Os n´umeros gaussianos de Fibonacci e relac¸ ˜oes recorrentes bidimensionais, Revista Thema, 16(4) (2019), 745–754. http://dx.doi.org/10.15536/thema.V16.2019.745-754.1333
[9] J. Chimpanzo, P. Catarino, P. Vasco, A. Borges, Bidimensional extensions of balancing and Lucas-balancing numbers, J. Discrete Math. Sci. Cryptogr., 27(1) (2024), 95–115.
[10] J. Chimpanzo, M. V. Otero-Espinar, A. Borges, P. Vasco, P. Catarino, Bidimensional extensions of cobalancing and Lucas-cobalancing numbers, Ann. Math. Sil., (2023), https://doi.org/10.2478/amsil-2023-0022

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

José Chimpanzo ^*
0009-0002-8658-5929
Portugal

Paula Maria Machado Cruz Catarino
0000-0001-6917-5093
Portugal

María Victoria Otero Espinar
0000-0002-0201-0523
Spain

Early Pub Date

March 30, 2024

Publication Date

May 23, 2024

Submission Date

November 20, 2023

Acceptance Date

March 24, 2024

Published in Issue

Year 2024 Volume: 7 Number: 2

DOI

https://doi.org/10.32323/ujma.1393569

IZ

https://izlik.org/JA55NW58LX

APA

Chimpanzo, J., Catarino, P. M. M. C., & Otero Espinar, M. V. (2024). Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences. Universal Journal of Mathematics and Applications, 7(2), 68-75. https://doi.org/10.32323/ujma.1393569

AMA

1.Chimpanzo J, Catarino PMMC, Otero Espinar MV. Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences. Univ. J. Math. Appl. 2024;7(2):68-75. doi:10.32323/ujma.1393569

Chicago

Chimpanzo, José, Paula Maria Machado Cruz Catarino, and María Victoria Otero Espinar. 2024. “Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences”. Universal Journal of Mathematics and Applications 7 (2): 68-75. https://doi.org/10.32323/ujma.1393569.

EndNote

Chimpanzo J, Catarino PMMC, Otero Espinar MV (May 1, 2024) Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences. Universal Journal of Mathematics and Applications 7 2 68–75.

IEEE

[1]J. Chimpanzo, P. M. M. C. Catarino, and M. V. Otero Espinar, “Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences”, Univ. J. Math. Appl., vol. 7, no. 2, pp. 68–75, May 2024, doi: 10.32323/ujma.1393569.

ISNAD

Chimpanzo, José - Catarino, Paula Maria Machado Cruz - Otero Espinar, María Victoria. “Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences”. Universal Journal of Mathematics and Applications 7/2 (May 1, 2024): 68-75. https://doi.org/10.32323/ujma.1393569.

JAMA

1.Chimpanzo J, Catarino PMMC, Otero Espinar MV. Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences. Univ. J. Math. Appl. 2024;7:68–75.

MLA

Chimpanzo, José, et al. “Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences”. Universal Journal of Mathematics and Applications, vol. 7, no. 2, May 2024, pp. 68-75, doi:10.32323/ujma.1393569.

Vancouver

1.José Chimpanzo, Paula Maria Machado Cruz Catarino, María Victoria Otero Espinar. Some Identities and Generating Functions for Bidimensional Balancing and Cobalancing Sequences. Univ. J. Math. Appl. 2024 May 1;7(2):68-75. doi:10.32323/ujma.1393569