# 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

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: ysimsek@akdeniz.edu.tr)

Abstract

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

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