Fibonacci Sums Modulo 5

Adegoke, Kunle; Frontczak, Robert; Goy, Taras

doi:10.2478/amsil-2024-0015

Abstract

We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.

References

K. Adegoke, Weighted sums of some second-order sequences, Fibonacci Quart. 56 (2018), no. 3, 252–262.
Search in Google Scholar Back to article
K. Adegoke, R. Frontczak, and T. Goy, New binomial Fibonacci sums, Palest. J. Math. 13 (2024), no. 1, 323–339.
Search in Google Scholar Back to article
K. Adegoke, R. Frontczak, and T. Goy, Binomial Fibonacci sums from Chebyshev polynomials, J. Integer Seq. 26 (2023), no. 9, Paper No. 23.9.6, 26 pp.
Search in Google Scholar Back to article
K. Adegoke, R. Frontczak, and T. Goy, On Fibonacci and Lucas binomial sums modulo 5, in: XIX International Scientific Mykhailo Kravchuk Conference. Abstracts, I. Sikorsky Kyiv Polytechnic Institute, Kyiv, 2023, pp. 60–61.
Search in Google Scholar Back to article
K. Adegoke, A. Olatinwo, and S. Ghosh, Cubic binomial Fibonacci sums, Electron. J. Math. 2 (2021), 44–51.
Search in Google Scholar Back to article
M. Bai, W. Chu, and D. Guo, Reciprocal formulae among Pell and Lucas polynomials, Mathematics 10 (2022), no. 15, 2691, 11 pp.
Search in Google Scholar Back to article
Z. Fan and W. Chu, Convolutions involving Chebyshev polynomials, Electron. J. Math. 3 (2022), 38–46.
Search in Google Scholar Back to article
R. Frontczak and T. Goy, Chebyshev-Fibonacci polynomial relations using generating functions, Integers 21 (2021), Paper No. A100, 15 pp.
Search in Google Scholar Back to article
I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, Elsevier/Academic Press, Amsterdam, 2015.
Search in Google Scholar Back to article
E. Hansen, Addendum for ‘A Table of Series and Products’, preprint (undated), 56 pp. Available at carmamaths.org/resources/jon/Preprints/Books/Other/hansen07.pdf.
Search in Google Scholar Back to article
E. Kılıç, S. Koparal, and N. Ömür, Powers sums of the first and second kinds of Chebyshev polynomials, Iran. J. Sci. Technol. Trans. A Sci. 44 (2020), no. 2, 425–435.
Search in Google Scholar Back to article
T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
Search in Google Scholar Back to article
D.J. Leeming, The coefficients of sinh xt/sin t and the Bernoulli polynomials, Internat. J. Math. Ed. Sci. Tech. 28 (1997), no. 4, 575–579.
Search in Google Scholar Back to article
C. Li and Z. Wenpeng, Chebyshev polynomials and their some interesting applications, Adv. Difference Equ. (2017), Paper No. 303, 9 pp.
Search in Google Scholar Back to article
Y. Li, On Chebyshev polynomials, Fibonacci polynomials, and their derivatives, J. Appl. Math. (2014), Art. ID 451953, 8 pp.
Search in Google Scholar Back to article
J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003.
Search in Google Scholar Back to article
N.J.A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences. Available at oeis.org/book.html.
Search in Google Scholar Back to article
S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section. Theory and Applications, Dover Publications, Inc., Mineola, NY, 2008.
Search in Google Scholar Back to article
W. Zhang, On Chebyshev polynomials and Fibonacci numbers, Fibonacci Quart. 40 (2002), no. 5, 424–428.
Search in Google Scholar Back to article

Fibonacci Sums Modulo 5

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