Integral representations for Mersenne and Horadam-Fermat numbers

Year 2024, Volume: 9 Issue: 3, 185 - 200, 31.12.2024

Ahmet İpek

https://doi.org/10.30931/jetas.1553048

Abstract

In this note we first derive integral representations for Mersenne numbers $M_{kn}$ and Horadam-Fermat numbers $\mathcal{F}_{kn}$, then we use those to provide integral representations for Mersenne numbers $M_{kn+r}$ and Horadam-Fermat numbers $\mathcal{F}_{kn+r}$, where $n\in\mathbb{Z}_{>0}=\{1,2,3,\ldots\}$ is a non-negative integer, $k\in\mathbb{Z}_{>0}=$ $\{1,2,3,\ldots\}$ is an arbitrary but fixed positive integer, while $r\in\mathbb{Z}_{\geqslant0}$ is an arbitrary but fixed non-negative integer.

Keywords

Mersenne number , Fermat number , Horadam number , Integral representations.

References

• [1] Andrica, D., Bagdasar, O., "Recurrent Sequences", 2020, Springer, Berlin.
• [2] Bernhart, F., "Catalan, Motzkin, and Riordan numbers", Discrete Math. 204 (1999) : 73-112.
• [3] Dana-Picard, T., "Parametric integrals and Catalan numbers", International Journal of Mathematical Education in Science and Technology 36(4) (2005) : 410-414.
• [4] Dana-Picard, T., Zeitoun., D.G., "Closed forms for 4-parameter families of integrals", International Journal of Mathematical Education in Science and Technology 40(6) (2009) : 828-837.
• [5] Dana-Picard, T., "Integral presentations of Catalan numbers", International Journal of Mathematical Education in Science and Technology 41(1) (2010) : 63-69.
• [6] Dana-Picard, T., "Integral presentations of Catalan numbers and Wallis formula", International Journal of Mathematical Education in Science and Technology 42(1) (2011) : 122-129.
• [7] Dana-Picard, T., Zeitoun, D.G., "Parametric improper integrals, Wallis formula and Catalan numbers", International Journal of Mathematical Education in Science and Technology 43(4) (2012) : 515-520.
• [8] Deza, E., "Mersenne numbers and Fermat numbers (Vol. 1)", World Scientific, Singapore, 2021.
• [9] Glasser, M.L., Zhou, Y., "An integral representation for the Fibonacci numbers and their generalization", Fibonacci Quart. 53( 4) (2015) : 313-318.
• [10] Grimaldi, R., "Fibonacci and Catalan Numbers: an introduction", John Wiley Sons, 2012.
• [11] Horadam, A.F., "Generating functions for powers of a certain generalised sequence of numbers", Duke Mathematical Journal 32(3) (1965) : 437-446.
• [12] Horadam, A.F., "Basic properties of a certain generalized sequence of numbers", The Fibonacci Quarterly 3 (1965) : 161-176.
• [13] Horadam, A.F., "Special properties of the sequence Wn(a; b; p; q)", The Fibonacci Quarterly 5 (1967) : 424-434.
• [14] Mccalla, P., Nkwanta, A., "Catalan and Motzkin integral representations. in The Golden Anniversary Celebration of the National Association of Mathematicians, Contemporary Mathematics", American Mathematical Society 759 (2020) : 125134.
• [15] Li, W., Cao, J., Niu, D., Zhao, J., Qi, F., "A brief survey and an analytic generalization of the Catalan numbers and their integral representations", Mathematics 11(8) (2023) : 1870.
• [16] Keskin, R., Siar, Z., "Some new identities concerning the Horadam sequence and its companion sequence", Communications of the Korean Mathematical Society 34(1) (2019) : 1-16.
• [17] Koshy, T., "Catalan numbers with applications", Oxford University Press, 2008.
• [18] Penson, K.A., Sixdeniers, J.M., "Integral representations of Catalan and related numbers", J. Integer Seq. 4(2) (2001) : Article 01.2.5.
• [19] Gi, F., Guo, B.N., "Integral representations of the Catalan numbers and their applications", Mathematics 5(3) (2017) : 40.
• [20]Stewart, S.M., "Simple integral representations for the Fibonacci and Lucas Numbers", Aust. J. Math. Anal. Appl. 19(2) (2022) : 2-5.