Generalized Formulas for Summation and Alternating Summation of Jacobsthal and Jacobsthal – Lucas Numbers

Mirwais Mansoor Kakar

doi:10.62810/jnsr.v4i2.377

Authors

Mirwais Mansoor Kakar Ghazni University, Mathematics Department, Faculty of Education, Ghazni, Afghanistan

DOI:

https://doi.org/10.62810/jnsr.v4i2.377

Keywords:

Generalized alternating summation formula, Generalized summation formula, Jacobsthal numbers, Jacobsthal -Lucas numbers

Abstract

Among integer sequences, the Jacobsthal and Jacobsthal–Lucas sequences are important sequences with applications in various areas of mathematics and applied sciences. Recent studies have examined these sequences, their properties, and summation formulas, including several generalizations. Motivated by these studies, this paper derives generalized summation and alternating summation formulas for the Jacobsthal and Jacobsthal–Lucas sequences. In particular, generalized summation formulas of the form sum from k = 1 to n of J_(mk+r) and sum from k = 1 to n of j_(mk+r) are established, where J_n and j_n denote the nth Jacobsthal and Jacobsthal–Lucas numbers, respectively, and m and r are integers with m not equal to 0. Generalized summation formulas with alternating signs are also derived, including sums of (-1)^(k-1)J_(mk+r) and (-1)^(k-1)j_(mk+r). For selected values of m and r, corresponding particular cases are presented for each generalized formula. In addition, summation and alternating summation formulas involving Jacobsthal and Jacobsthal–Lucas numbers with negative indices are derived, including sums involving J_(-k), j_(-k), (-1)^(k-1)J_(-k), and (-1)^(k-1)j_(-k), together with several related formulas.

Downloads

Download data is not yet available.

References

Babadag, F., Mansoor Kakar, M., & Atasoy, A. (2024). A New Approach to Dual Jacobsthal Split Quaternions with Different Polar Representation. Journal of Advances in Mathematics and Computer Science, 39(2):52-62. https://10.9734/JAMCS/2024/v39i21867 DOI: https://doi.org/10.9734/jamcs/2024/v39i21867

Bender, E. A., & Goldman, J. R. (1971). Enumerative uses of generating functions. Indiana University Mathematics Journal, 20, 753 - 764. https://oeis.org/A000110/a000110_5.pdf DOI: https://doi.org/10.1512/iumj.1971.20.20060

Bicknell, M. (1975). A primer of the Pell sequence and related sequences. The Fibonacci Quarterly, 13 (4), 345 – 349. https://www.fq.math.ca/Scanned/13-4/bicknell.pdf DOI: https://doi.org/10.1080/00150517.1975.12430627

Bród, D., & Michalski, A., (2022). On generalized Jacobsthal and Jacobsthal - Lucas numbers. Annales Mathematicae Silesianae, 36(2). https:// 10.2478/amsil-2022-0011 DOI: https://doi.org/10.2478/amsil-2022-0011

Brousseau, B. A. (1968). A sequence of power formulas. The Fibonacci Quarterly, 6 (1), 81 – 83. https://www.fq.math.ca/Scanned/6-1/brousseau3.pdf DOI: https://doi.org/10.1080/00150517.1968.12431264

Campos, H., Catarino, P., Aires, A. P., Vasco, P., & Borges, A. (2014). On Some Identities of k-Jacobsthal-Lucas Numbers. Int. Journal of Math. Analysis, 8 (10), 489 – 494. https://www.m-hikari.com/ijma/ijma-2014/ijma-9-12-2014/catarinoIJMA9-12-2014.pdf DOI: https://doi.org/10.12988/ijma.2014.4249

Catarino, P., Vasco, P., Campos, H., Aires, A. P., & Borges, A. (2015). New families of Jacobsthal and Jacobsthal–Lucas numbers. Algebra and Discrete Mathematics, 20 (1), 40 – 54. http://nbuv.gov.ua/UJRN/Adm_2015_20_1_7

Dasdemir, A. (2019). Mersenne, Jacobsthal, and Jacobsthal–Lucas numbers with negative subscripts. Acta Mathematica Universitatis Comenianae, 88 (1), 145 - 156. https://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/906/645

Frontczak, R. (2018). Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach. Notes on Number Theory and Discrete Mathematics, 24 (2), 94 - 103. https://10.7546/nntdm.2018.24.2.94-103 DOI: https://doi.org/10.7546/nntdm.2018.24.2.94-103

Graham, R. L., Knuth, D. E., & Oren, P. (1989). Concrete Mathematics. Reading, MA: Addison–Wesley.

Hoggatt, V. E., Jr., & Bicknell-Johnson, M. (1978). Convolution arrays for Jacobsthal and Fibonacci polynomials. The Fibonacci Quarterly, 16 (5), 385 - 402. https://www.fq.math.ca/Scanned/16-5/hoggatt1.pdf DOI: https://doi.org/10.1080/00150517.1978.12430290

Horadam, A. F. (1961). A generalized Fibonacci sequence. American Mathematical Monthly, 68 (5), 455 - 459. http://www.jstor.org/stable/2311099 DOI: https://doi.org/10.1080/00029890.1961.11989696

Horadam, A. F. (1961). Fibonacci number triples. American Mathematical Monthly, 68 (8), 751 - 753. https://doi.org/10.1080/00029890.1961.11989762 DOI: https://doi.org/10.1080/00029890.1961.11989762

Horadam, A. F. (1988). Jacobsthal and Pell curves. The Fibonacci Quarterly, 26 (1), 79 - 83. https://www.fq.math.ca/Scanned/26-1/horadam2.pdf DOI: https://doi.org/10.1080/00150517.1988.12429664

Horadam, A. F. (1996). Jacobsthal representation numbers. The Fibonacci Quarterly, 34, 40 - 54. https://doi.org/10.1080/00150517.1996.12429096 DOI: https://doi.org/10.1080/00150517.1996.12429096

Horadam, A. F. (1997). Jacobsthal representation polynomials. The Fibonacci Quarterly, 35, 137 - 148. https://doi.org/10.1080/00150517.1997.12429009 DOI: https://doi.org/10.1080/00150517.1997.12429009

Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3, 161 - 176. https://doi.org/10.1080/00150517.1965.12431416 DOI: https://doi.org/10.1080/00150517.1965.12431416

Jhala, D., Sisodiya, K., & Rathore, G. P. S. (2013). On Some Identities for k-Jacobsthal Numbers. Int. Journal of Math. Analysis, 7 (12), 551 – 556. http://dx.doi.org/10.12988/ ijma.2014.4249 DOI: https://doi.org/10.12988/ijma.2013.13052

Jhala, D., Rathore, G. P. S., & Sisodiya, K. (2014). Some properties of k–Jacobsthal numbers with arithmetic indexes. Turkish Journal of Analysis and Number Theory, 2 (4), 119 - 124. http://10.12691/TJANT-2-4-3 DOI: https://doi.org/10.12691/tjant-2-4-3

Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. A Wiley-Interscience Publication. DOI: https://doi.org/10.1002/9781118033067

Mansoor Kakar, M., & Mehrad,A. A. (2025). A new approach to hyper dual numbers with tribonacci and tribonacci-Lucas numbers and their generalized summation formulas. Journal of Innovative Research in Mathematical and Computational Sciences, 5(1):66-81.https:// 10.58205/jiamcs.v5i1.1885

Melham, R. (1999). Sums involving Fibonacci and Pell numbers. Portugalie Mathematica, 56 (3), 309 - 317. https://eudml.org/doc/48511

Silvester, J. R. (1979). Fibonacci properties by matrix methods. The Mathematical Gazette, 63 (425), 188 - 191. 10.2307/3617892 DOI: https://doi.org/10.2307/3617892

Sloane, N. J. A. (1964). The On-Line Encyclopedia of Integer Sequences.

Subba Rao, K. (1935). Some properties of Fibonacci numbers. American Mathematical Monthly, 60 (10), 680 - 684. 10.1080/00029890.1953.11988390 DOI: https://doi.org/10.1080/00029890.1953.11988390

Uygun, U. (2019). On the bounds for the norms of Toeplitz matrices with the Jacobsthal and Jacobsthal–Lucas numbers. Journal of Engineering Technology and Applied Sciences, 4 (3), 105 - 114. 10.30931/jetas.569742 DOI: https://doi.org/10.30931/jetas.569742

Wilf, H. S. (2006). Generating functionology (3rd ed.). Wellesley, MA: A K Peters, Ltd.