Parameterized families of polylog integrals

Year 2021, , 400 - 419, 13.12.2021

Anthony Sofo , Necdet Batir

https://doi.org/10.33205/cma.1006384

Cited By: 1

Abstract

It is commonly known that integrals containing log-polylog integrands admit representations in terms
of special functions such as the Dirichlet eta and Dirichlet beta functions. We investigate two parameterized families
of such integrals and in a particular case demonstrate a connection with the Herglotz function. In the process of the
investigation we recover some known Euler sum equalities and discover some new identities.

Keywords

Harmonic number, gamma function, alternating harmonic number, Riemann zeta function, polylogarithm function, polygamma functions, linear Euler sums

Supporting Institution

none

Project Number

none

Thanks

Thank you

References

• H. Alzer, J. Choi: Four parametric linear Euler sums, J. Math. Anal. Appl., 484 (1) (2020), 123661, 22 pp.
• H. Alzer, A. Sofo: New series representations for Apéry’s and other classical constants, Anal. Math., 44 (3) (2018), 287–297.
• K. C. Au: Evaluation of one dimensional polylogarithmic integral with applications to infinite series, arXiv:2007.03957v2. (2020).
• N. Batır: On some combinatorial identities and harmonic sums, Int. J. Number Theory, 13 (7) (2017), 1695–1709.
• G. Boros, V. Moll: Irresistible integrals. Symbolics, analysis and experiments in the evaluation of integrals, Cambridge University Press, Cambridge, xiv+306 pp. ISBN: 0-521-79636-9, (2004).
• M. W. Coffey: Evaluation of a ln tan integral arising in quantum field theory, J. Math. Phys., 49 (9), 093508, (2008),15 pp.
• M.W. Coffey: Some definite logarithmic integrals from Euler sums, and other integration results, arXiv:1001.1366. (2010).
• H. Cohen: Number Theory. Analytic and Modern Tools. Graduate Texts in Mathematics, vol. II. Springer, New York, (2007).
• R. Crandall: Unified algorithms for polylogarithm, L-series, and zeta variants, Algorithmic Reflections: Selected Works. PSI press, www.marvinrayburns.com/UniversalTOC25.pdf. (2012).
• A. Dixit, R. Gupta and R. Kumar: Extended higher Herglotz functions I. Functional equations, arXiv:2107.02607v1. (2021).
• A. Erdélyi, W. Magnus, F. Oberhettinger and F. G.Tricomi: Higher Transcendental Functions, Vol. 1. New York: Krieger, (1981).
• S. R. Finch: Mathematical constants. II. Encyclopedia of Mathematics and its Applications, 169. Cambridge University Press, Cambridge, xii+769 pp. ISBN: 978-1-108-47059-9, (2019).
• P. Flajolet, B. Salvy: Euler sums and contour integral representations, Experiment. Math., 7 (1), (1998), 15–35.
• P. Freitas: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comp., 74 (251) (2005), 1425–1440.
• G. Herglotz: Uber die Kroneckersche Grenzformel fur reelle, quadratische Korper I, Ber. Verhandl. Sachsischen Akad. Wiss. Leipzig 75, pp. 3–14 (1923).
• R. Lewin: Polylogarithms and Associated Functions, North Holland, New York, (1981).
• I. Mez˝o: Log-sine-polylog integrals and alternating Euler sums, Acta Math. Hungar., 160 (1) (2020), 45–57.
• V. Moll: Special integrals of Gradshteyn and Ryzhik—the proofs. Vol. II, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, xi+263 pp. ISBN: 978-1-4822-5653-6, (2016).
• H. Muzaffar, K. S. Williams: A restricted Epstein zeta function and the evaluation of some definite integrals, Acta Arith., 104 (1) (2002), 23–66.
• P. Nahin: Inside interesting integrals (with an introduction to contour integration), Second edition. Undergraduate Lecture Notes in Physics. Springer, Cham, [2020], c . xlvii+503 pp. ISBN: 978-3-030-43787-9; 978-3-030-43788-6, (2020).
• N. Nielsen: Die Gammafunktion, Chelsea Publishing Company, Bronx and New York, (1965).
• D. Radchenko, D. Zagier: Arithmetic properties of the Herglotz function, arXiv:2012.15805v1., (2020).
• A. Sofo: Integrals of polylogarithmic functions with alternating argument, Asian-Eur. J. Math., 13 (7) 14 pp. (2020).
• A. Sofo: Integral identities for sums, Math. Commun, 13 (2) (2008), 303–309.
• A. Sofo, H. M. Srivastava: A family of shifted harmonic sums, Ramanujan J., 37 (1) (2015), 89–108.
• A. Sofo: New classes of harmonic number identities, J. Integer Seq., 15 (7) (2012) Article 12.7.4, 12 pp.
• A. Sofo, D. Cvijovi´c: Extensions of Euler harmonic sums, Appl. Anal. Discrete Math., 6 (2) (2012), 317–328.
• A. Sofo: Shifted harmonic sums of order two, Commun. Korean Math. Soc., 29 (2) (2014), 239–255.
• A. Sofo: General order Euler sums with rational argument, Integral Transforms Spec. Funct., 30 (12) (2019), 978–991.
• A. Sofo: General order Euler sums with multiple argument, J. Number Theory, 189 (2018). 255–271.
• A. Sofo: Evaluation of integrals with hypergeometric and logarithmic functions, Open Math., 16 (1) (2018), 63–74.
• A. Sofo, A. S. Nimbran: Euler-like sums via powers of log, arctan and arctanh functions, Integral Transforms Spec. Funct., 31 (12) (2020), 966–981.
• H. M. Srivastava, J. Choi: Zeta and q-Zeta functions and associated series and integrals. Elsevier, Inc., Amsterdam, 2012. xvi+657 pp. ISBN: 978-0-12-385218-2.
• H. M. Srivastava, J. Choi: Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, x+388 pp. ISBN: 0-7923-7054-6 (2001).
• C. I. V˘alean: (Almost) impossible integrals, sums, and series. Problem Books in Mathematics, Springer, Cham, xxxviii+539 pp. ISBN: 978-3-030-02461-1; 978-3-030-02462-8 41-01, (2019).
• C. Xu, Y. Yan and Z. Shi: Euler sums and integrals of polylogarithm functions, J. Number Theory, 165 (2016), 84–108.
• D. Zagier: A Kronecker limit formula for real quadratic fields, Math. Ann.,213 (1975), 153–184 .