Article ID: MTJPAM-D-21-00002

Title: Some Classes of Finite Sums Related to the Generalized Harmonic Functions and Special Numbers and Polynomials

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00002; Volume 4 / Issue 3 / Year 2022 (Special Issue), Pages 61-79

Document Type: Research Paper

Author(s): Yilmaz Simsek a

aDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya-TURKEY

Received: 3 January 2021, Accepted: 11 September 2021, Published: 12 October 2021.

Corresponding Author: Yilmaz Simsek (Email address:

Full Text: PDF


The aim of this paper is to give some new classes of finite sums involving the numbers y(m, λ), the generalized harmonic functions, special numbers and polynomials, the Dedekind sums, and other combinatorial sum. Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.

Keywords: Generating functions, Special numbers and polynomials, Finite sum, Generalized harmonic functions, Riemann zeta function, Hurwitz zeta function, Hypergeometric series, Polylogarithms, Digamma functions, Polygmamma functions, Dedekind sums, Hardy sums

  1. H. Alzer and J. Choi, The Riemann zeta function and classes of infinite series, Appl. Anal. Discrete Math. 11, 386–398, 2017.
  2. T. M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17, 147–157, 1950.
  3. T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1, 161–167, 1951.
  4. T. M. Apostol, Modular functions and Dirichlet series in number theory, Springer-Verlag, New York, 1976.
  5. T.-T. Bai and Q.-M. Luo, A Simple proof of a binomial identity with applications, Montes Taurus J. Pure Appl. Math. 1 (2), 13–20, 2019; Article ID: MTJPAM-D-19-00008.
  6. A. Bayad, Jacobi forms in two variables: multiple elliptic Dedekind sums, the Kummer-Von Staudt Clausen congruences for elliptic Bernoulli functions and values of Hecke L-functions, Montes Taurus J. Pure Appl. Math. 1 (2), 58–129, 2019; Article ID: MTJPAM-D-19-00009.
  7. M. Beck, Geometric proofs of polynomial reciprocity laws of Carlitz, Berndt, and Dieter. In: Diophantine Analysis and Related Fields. Sem. Math. Sci. 35, 11–18, 2006.
  8. F. Beukers, Chapter 2: Gauss’ Hypergeometric Function, In: Arithmetic and geometry around hypergeometric functions (Edited by Rolf-Peter Holzapfel, Muhammed Uludag, M. Yoshida), Lecture Notes of a CIMPA Summer School held at Galatasaray University, Istanbul,Part of the Progress in Mathematics book series (PM, volume 260), Birkhäuser Verlag Basel/Switzerland, pp. 23–42, 2007.
  9. R. E. Bradley and C. E. Sandifer, Leonhard Euler life, work and legacy, Elsevier Science, Amsterdam, 2007.
  10. M. C. Boztas and M. Can, Transformation formulas of a character analogue of logθ2(z), Ramanujan J. 48, 323–349, 2019.
  11. M. Can, Reciprocity formulas for Hall–Wilson–Zagier type Hardy–Berndt sums, Acta Math. Hungar. 163 (1), 118–139, 2021.
  12. E. Cetin, Y. Simsek and I. N. Cangul, Some special finite sums related to the three-term polynomial relations and their applications, Advanc. Differ. Equ. 2014, 2014:283;
  13. J. Choi, Remark on the Hurwitz-Lerch zeta function, Fixed Point Theory and Appl., 2013 (70), 2013.
  14. J. Choi and H. M. Srivastava, Certain families of series associated with the Hurwitz-Lerch Zeta function, Appl. Math. Comput. 170 (1), 399–409, 2005.
  15. J. Choi, Certain summation formulas involving harmonic numbers and generalized harmonic numbers, Appl. Math. Comput. 218 (3), 734–740, 2011; doi:10.1016/j.amc.2011.01.062
  16. J. Choi and H.M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Mathematical and Computer Modelling 54 (910), 2220–2234, 2011.
  17. W. Chu and L. De Donno, Hypergeometric series and harmonic number identities, Advances in Applied Mathematics 34, 123–137, 2005.
  18. L. Comtet, Advanced combinatorics. Dordrecht-Holland/ Boston-U.S.A.: D. Reidel Publication Company, 1974.
  19. J. H. Conway and R. K. Guy, The book of numbers, Springer-Verlag, New York, NY, USA, 1996.
  20. M. C. Dağli and M. Can, A new generalization of Hardy–Berndt sums, Proc. Indian Acad. Sci. (Math. Sci.) 123, 177–192, 2013.
  21. A. Dil, L. Mezo and M. Cenkci, Evaluation of Euler-like sums via Hurwitz zeta values, Turk. J. Math. 41, 1640–1655, 2017.
  22. L. A. Goldberg, Transformation of theta functions and analogues of Dedekind sums, Thesis, University of Illinois Urbana, 1981.
  23. R. Golombek, Aufgabe 1088, El. Math. 49, 126–127, 1994.
  24. R. Golombek, D. Marburg, Aufgabe 1088, Summen mit Quadraten von Binomialkoeffizienten, El. Math. 50, 125–131, 1995.
  25. G. H. Hardy, On certain series of discontinuous functions connected with the modular functions, Q. J. Math. 36, 93-123, 1905 (= Collected Papers, Vol. IV Clarendon Press, Oxford, pp. 362–392, 1969).
  26. T. Kim, D. S. Kim and J. Kwon, Analogues of Faulhaber’s formula for poly-Bernoulli and type 2 poly-Bernoulli polynomials, Montes Taurus J. Pure Appl. Math., 3 (1), 1–6, 2021; Article ID: MTJPAM-D-20-00033
  27. W. Koepf, Hypergeometric summation, an algorithmic approach to summation and special function identities, Second edition, Springer-Verlag, London, 2014.
  28. I. Kucukoglu and Y. Simsek, Identities and relations on the q-Apostol type Frobenius-Euler numbers and polynomials, J. Korean Math. Soc. 56 (1), 265–284, 2019.
  29. H. Maier and M. Th. Rassias, The maximum of cotangent sums related to Estermann’s zeta function in rational numbers in short intervals, Appl. Anal. Discrete Math. 11, 166–176, 2017.
  30. G. V. Milovanovic and Y. Simsek, Dedekind and Hardy type sums and trigonometric sums induced by quadrature formulas, In: Trigonometric Sums and Their Applications. eBook ISBN 978-3-030-37904-9, Raigorodskii, Andrei M., Rassias, Michael Th. (Eds.), Springer Nature Publishing Group Switzerland, Basel, pp. 183–228, 2020.
  31. V. H. Moll, Numbers and functions: from a classical-experimental mathematician’s point of view, Student Mathematical Library, vol.65, American Mathematical Society, Providence, Rhode Island, 2012.
  32. H. Rademacher and E. Grosswald, Dedekind sums, Carus Mathematical Monograph, No. 16, Mathematical Association of America, Washington D.C., 1972.
  33. M. Th. Rassias, A cotangent sum related to zeros of the Estermann zeta function, Appl. Math. Comput. 240, 161–167, 2014.
  34. T. M. Rassias and H. M. Srivastava, Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput. 131, 734–740, 2011.
  35. S.-H. Rim, T. Kim and S. S. Pyo, Identities between harmonic, hyperharmonic and Daehee numbers, J. Inequal Appl. 1, 168, 2018.
  36. C. E. Sandifer, How Euler Did Even More, The Mathematical Association of America, Washington, 2014.
  37. Y. Simsek, Generalized Dedekind sums associated with the Abel sum and the Eisenstein and Lambert series. Adv. Stud. Contemp. Math. 9, 125–137, 2004.
  38. Y. Simsek, Remarks on reciprocity laws of the Dedekind sums and Hardy sums, Adv. Stud. Contemp. Math. (Kyungshang) 12 (2), 237–246, 2006.
  39. Y. Simsek, On Analytic properties and character analogs of Hardy sums, Taiwanese J. Math. 13 (1), 253–268, 2009.
  40. Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys. 17 (4), 495–508, 2010.
  41. Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12, 1–35, 2018;
  42. Y. Simsek, Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math. 1 (1), 1–76, 2019; Article ID: MTJPAM-D-19-00005.
  43. Y. Simsek, Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers, J. Math. Anal. Appl. 477, 1328–1352, 2019.
  44. Y. Simsek, Interpolation functions for new classes special numbers and polynomials via applications of p-adic integrals and derivative operator, Montes Taurus J. Pure Appl. Math. 3 (1), 38–61, 2011.
  45. Y. Simsek, Miscellaneous formulae for the certain class of combinatorial sums and special numbers. Bulletin T.CLIV de l’Académie serbe des sciences et des arts 46, 151–167, 2021.
  46. Y. Simsek, New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums, RACSAM 115 (66), 2021;
  47. A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory 154, 144–159, 2015.
  48. A. Sofo and H. M. Srivastava , Identities for the harmonic numbers and binomial coefficients, The Ramanujan Journal 25, 93–113, 2011.
  49. H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129, 77–84, 2000.
  50. H. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  51. H. M. Srivastava and J. Choi, Zeta and q -zeta functions and associated series and integrals, Amsterdam, Elsevier Science Publishers, 2012.
  52. C. I. Valean, (Almost) Impossible integrals, sums, and series, Springer International Publishing, 2019.
  53. A. Xu, On an open problem of Simsek concerning the computation of a family of special numbers, Appl. Anal. Discrete Math. 13 (1), 61–72, 2019;