Families of skew linear harmonic Euler sums involving some parameters

Anthony Sofo

doi:10.56754/0719-0646.2601.075

DOI:

https://doi.org/10.56754/0719-0646.2601.075

Abstract

In this study we investigate a family of skew linear harmonic Euler sums involving some free parameters. Our analysis involves using the properties of the polylogarithm function, commonly referred to as the Bose-Einstein integral. A reciprocity property is utilized to highlight an explicit representation for a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature and introduces some new results.

Keywords

Skew linear harmonic Euler sum , Polygamma function , harmonic number , polylogarithm function , Bernoulli number

Mathematics Subject Classification:

11M06 , 11M35 , 26B15 , 33B15 , 42A70 , 65B10

Anthony Sofo College of Engineering and Science, Victoria University, Australia. https://orcid.org/0000-0002-1277-8296

Pages: 75–89
Date Published: 2024-04-05
Vol. 26 No. 1 (2024)

H. Alzer and J. Choi, “Four parametric linear Euler sums,” J. Math. Anal. Appl., vol. 484, no. 1, 2020, Art. ID 123661, doi: 10.1016/j.jmaa.2019.123661.

D. Borwein, J. M. Borwein, and R. Girgensohn, “Explicit evaluation of Euler sums,” Proc. Edinburgh Math. Soc. (2), vol. 38, no. 2, pp. 277–294, 1995, doi: 10.1017/S0013091500019088.

J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer, “Central binomial sums, multiple Clausen values, and zeta values,” Experiment. Math., vol. 10, no. 1, pp. 25–34, 2001, doi: 10.1080/10586458.2001.10504426.

W. Chu, “Infinite series on quadratic skew harmonic numbers,” Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, vol. 117, no. 2, 2023, Art. ID 75, doi: 10.1007/s13398-023- 01407-9.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vols. I, II. McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953.

P. Flajolet and B. Salvy, “Euler sums and contour integral representations,” Experiment. Math., vol. 7, no. 1, pp. 15–35, 1998, doi: 10.1080/10586458.1998.10504356.

L. Lewin, Polylogarithms and associated functions. North-Holland Publishing Co., New York- Amsterdam, 1981.

L. A. Medina and V. H. Moll, “The integrals in Gradshteyn and Ryzhik part 27: More logarithmic examples,” Scientia, vol. 26, pp. 31–47, 2015.

N. Nielsen, Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten. Chelsea Publishing Co., New York, 1965.

A. S. Nimbran, P. Levrie, and A. Sofo, “Harmonic-binomial Euler-like sums via expansions of (arcsinx)p,” Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, vol. 116, no. 1, 2022, Art. ID 23, doi: 10.1007/s13398-021-01156-7.

A. Sofo, “General order Euler sums with multiple argument,” J. Number Theory, vol. 189, pp. 255–271, 2018, doi: 10.1016/j.jnt.2017.12.006.

A. Sofo, “General order Euler sums with rational argument,” Integral Transforms Spec. Funct., vol. 30, no. 12, pp. 978–991, 2019, doi: 10.1080/10652469.2019.1643851.

A. Sofo and J. Choi, “Extension of the four Euler sums being linear with parameters and series involving the zeta functions,” J. Math. Anal. Appl., vol. 515, no. 1, 2022, Art. ID 126370, doi: 10.1016/j.jmaa.2022.126370.

A. Sofo and A. S. Nimbran, “Euler-like sums via powers of log, arctan and arctanh functions,” Integral Transforms Spec. Funct., vol. 31, no. 12, pp. 966–981, 2020, doi: 10.1080/10652469.2020.1765775.

H. M. Srivastava, M. A. Chaudhry, A. Qadir, and A. Tassaddiq, “Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions,” Russ. J. Math. Phys., vol. 18, no. 1, pp. 107–121, 2011, doi: 10.1134/S1061920811010110.

H. M. Srivastava and J. Choi, Series associated with the zeta and related functions. Kluwer Academic Publishers, Dordrecht, 2001, doi: 10.1007/978-94-015-9672-5.

H. M. Srivastava and J. Choi, Zeta and q-Zeta functions and associated series and integrals. Elsevier, Inc., Amsterdam, 2012, doi: 10.1016/B978-0-12-385218-2.00001-3.

S. M. Stewart, “Explicit expressions for some linear Euler-type sums containing harmonic and skew-harmonic numbers,” J. Class. Anal., vol. 20, no. 2, pp. 79–101, 2022, doi: 10.7153/jca- 2022-20-07.