Identities for a Special Finite Sum Related to the Dedekind Sums and Fibonacci Numbers

Year 2023, Volume: 10 Issue: 2, 232 - 240, 27.06.2023

Elif Çetin

https://doi.org/10.54287/gujsa.1280707

Abstract

The origin of this article is to achieve original equations related to the special finite sum C(μ,β;1), which is connected with Dedekind, Hardy, Simsek, and many other finite sums. By using the analytic properties of this sum, many useful identities are established between the C(μ,β;1) sum and other well-known finite sums. Through the use of these identities, the reciprocity law of this sum is obtained. Furthermore, another reciprocity law of the sum C(μ,β;1) is presented for μ and β are particular Fibonacci numbers. This remarkable result establishes a connection between number theory and analysis.

Keywords

Hardy Sums , Dedekind Sums , Simsek Sums , Fibonacci Numbers

References

• Apostol, T. M. (1976). Modular functions and Dirichlet Series in Number Theory. Springer-Verlag.
• Apostol, T. M., & Vu, T. H. (1982). Elementary proofs of Berndt’s reciprocity laws. Pacific Journal of Mathematics, 98(1), 17-23. doi:10.2140/pjm.1982.98.17
• Berndt, B. C., & Dieter, U. (1982). Sums involving the greatest integer function and Riemann Stieltjes integration. Journal für die Reine und Angewandte Mathematik, 337, 208-220. doi:10.1515/crll.1982.337.208
• Berndt, B. C., & Goldberg, L. A. (1984). Analytic properties of arithmetic sums arising in the theory of the classical theta-functions. SIAM Journal on Mathematical Analysis, 15(1), 143-150. doi:10.1137/0515011
• Cetin, E., Simsek, Y., & Cangul, İ. N. (2014). Some Special Finite Sums Related to the Three-Term Polynomial Relations and Their Applications. Advances in Difference Equations, 2014, 283. doi:10.1186/1687-1847-2014-283
• Cetin, E. (2016a). A Note on Hardy Type Sums and Dedekind Sums. Filomat, 30(4), 977-983. doi:10.2298/FIL1604977C
• Cetin, E. (2016b). Analytic Properties of the Sum B_1 (h,k). Mathematical and Computational Applications, 21(3), 31. doi:10.3390/mca21030031
• Cetin, E. (2018, October 26-29). Remarks on Special Sums Associated with Hardy Sums. In: Proceedings of the Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2018), (pp. 153-156). Antalya.
• Cetin, E. (2023, March 11-13). A Note on Trigonometric ldentities of the Special Finite Sums. In: Proceedings of the 13th Symposium on Generating Functions of Special Numbers and Polynomials and their Applications (GFSNP 2023), which is dedicated to Professor Yilmaz Simsek on the Occasion of his 60th Anniversary, (pp. 1-7). Antalya.
• Dedekind, R. (1892). Erläuterungen zu zwei Fragmenten von Riemann-Riemann’s Gesammelte Math. Werke.
• Goldberg, L. A. (1981) Transformation of Theta-functions and analogues of Dedekind sums. MSc Thesis, University of Illinois.
• Hardy, G. H. (1905). On certain series of discontinues functions connected with the modular functions, Quart. J. Math., 36, 93-123.
• Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. John Wiley and Sons, New York, USA.
• Meyer, J. L. (2005). Symmetric Arguments in the Dedekind Sum. Fibonacci Quarterly, 43(2), 122.
• Milovanović, G. V., & Simsek, Y. (2020). Dedekind and Hardy Type Sums and Trigonometric Sums Induced by Quadrature Formulas. In: A. Raigorodskii, M. Rassias (Eds.), Trigonometric Sums and Their Applications (pp. 183-228). Springer, Cham. doi:10.1007/978-3-030-37904-9_10
• Pettet, M. R., & Sitaramachandrarao, R. (1987). Three-term relations for Hardy sums. Journal of Number Theory, 25(3), 328-339. doi:10.1016/0022-314X(87)90036-9
• Rademacher, H., & Grosswald, E. (1972). Dedekind sums. Carus Mathematical Monographs, The Mathematical Association of America.
• Sitaramachandrarao, R. (1987). Dedekind and Hardy sums. Acta Arithmetica, 48(4), 325-340. doi:10.4064/aa-48-4-325-340
• Simsek, Y. (1993). A note on Dedekind sums. Bull. Calcutta Math. Soc., 85(6), 567-572.
• Simsek, Y. (1998). Theorems on Three-Term Relations for Hardy Sum. Turkish Journal of Mathematics, 22(2), 153-162.
• Simsek, Y. (2003). Relation between theta-function Hardy sums Eisenstein and Lambert series in the transformation formula of logɳ g,h(z). Journal of Number Theory, 99(2) 338-360. doi:10.1016/s0022-314x(02)00072-0
• Simsek, Y. (2004). On generalized Hardy sums s_5 (h,k). Ukrainian Mathematical Journal, 56(10), 1712-1719. doi:10.1007/s11253-005-0146-2
• Simsek, Y. (2006). p -adic q-higher-order hardy-type sums. Journal of the Korean Mathematical Society, 43(1), 111-131.
• Simsek, Y. (2009a). q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions. Nonlinear Analysis, 71(12), e377-e395. doi:10.1016/j.na.2008.11.014
• Simsek, Y. (2009b). On analytic properties and character analogs of Hardy sums. Taiwanese Journal of Mathematics, 13(1), 253-268. doi:10.11650/twjm/1500405282
• Simsek, Y. (2010). Special functions related to Dedekind-type DC-sums and their applications. Russian Journal of Mathematical Physics, 17(4), 495-508. doi:10.1134/S1061920810040114
• Simsek, Y. (2022). Some classes of finite sums related to the generalized Harmonic functions and special numbers and polynomials. Montes Taurus Journal of Pure and Applied Mathematics, 4(3), 61-79.
• Simsek, Y. (2023). Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 117(3), 130. doi:10.1007/s13398-023-01464-0