Parameterized families of polylog integrals

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

References

1. H. Alzer, J. Choi: Four parametric linear Euler sums, J. Math. Anal. Appl., 484 (1) (2020), 123661, 22 pp.
2. H. Alzer, A. Sofo: New series representations for Apéry’s and other classical constants, Anal. Math., 44 (3) (2018), 287–297.
3. K. C. Au: Evaluation of one dimensional polylogarithmic integral with applications to infinite series, arXiv:2007.03957v2. (2020).
4. N. Batır: On some combinatorial identities and harmonic sums, Int. J. Number Theory, 13 (7) (2017), 1695–1709.
5. 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).
6. M. W. Coffey: Evaluation of a ln tan integral arising in quantum field theory, J. Math. Phys., 49 (9), 093508, (2008),15 pp.
7. M.W. Coffey: Some definite logarithmic integrals from Euler sums, and other integration results, arXiv:1001.1366. (2010).
8. H. Cohen: Number Theory. Analytic and Modern Tools. Graduate Texts in Mathematics, vol. II. Springer, New York, (2007).

9. R. Crandall: Unified algorithms for polylogarithm, L-series, and zeta variants, Algorithmic Reflections: Selected Works. PSI press, www.marvinrayburns.com/UniversalTOC25.pdf. (2012).
10. A. Dixit, R. Gupta and R. Kumar: Extended higher Herglotz functions I. Functional equations, arXiv:2107.02607v1. (2021).
11. A. Erdélyi, W. Magnus, F. Oberhettinger and F. G.Tricomi: Higher Transcendental Functions, Vol. 1. New York: Krieger, (1981).
12. 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).
13. P. Flajolet, B. Salvy: Euler sums and contour integral representations, Experiment. Math., 7 (1), (1998), 15–35.
14. P. Freitas: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comp., 74 (251) (2005), 1425–1440.
15. G. Herglotz: Uber die Kroneckersche Grenzformel fur reelle, quadratische Korper I, Ber. Verhandl. Sachsischen Akad. Wiss. Leipzig 75, pp. 3–14 (1923).
16. R. Lewin: Polylogarithms and Associated Functions, North Holland, New York, (1981).
17. I. Mez˝o: Log-sine-polylog integrals and alternating Euler sums, Acta Math. Hungar., 160 (1) (2020), 45–57.
18. 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).
19. H. Muzaffar, K. S. Williams: A restricted Epstein zeta function and the evaluation of some definite integrals, Acta Arith., 104 (1) (2002), 23–66.
20. 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).
21. N. Nielsen: Die Gammafunktion, Chelsea Publishing Company, Bronx and New York, (1965).
22. D. Radchenko, D. Zagier: Arithmetic properties of the Herglotz function, arXiv:2012.15805v1., (2020).
23. A. Sofo: Integrals of polylogarithmic functions with alternating argument, Asian-Eur. J. Math., 13 (7) 14 pp. (2020).
24. A. Sofo: Integral identities for sums, Math. Commun, 13 (2) (2008), 303–309.
25. A. Sofo, H. M. Srivastava: A family of shifted harmonic sums, Ramanujan J., 37 (1) (2015), 89–108.
26. A. Sofo: New classes of harmonic number identities, J. Integer Seq., 15 (7) (2012) Article 12.7.4, 12 pp.
27. A. Sofo, D. Cvijovi´c: Extensions of Euler harmonic sums, Appl. Anal. Discrete Math., 6 (2) (2012), 317–328.
28. A. Sofo: Shifted harmonic sums of order two, Commun. Korean Math. Soc., 29 (2) (2014), 239–255.
29. A. Sofo: General order Euler sums with rational argument, Integral Transforms Spec. Funct., 30 (12) (2019), 978–991.
30. A. Sofo: General order Euler sums with multiple argument, J. Number Theory, 189 (2018). 255–271.
31. A. Sofo: Evaluation of integrals with hypergeometric and logarithmic functions, Open Math., 16 (1) (2018), 63–74.
32. 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.
33. 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.
34. 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).
35. 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).
36. C. Xu, Y. Yan and Z. Shi: Euler sums and integrals of polylogarithm functions, J. Number Theory, 165 (2016), 84–108.
37. D. Zagier: A Kronecker limit formula for real quadratic fields, Math. Ann.,213 (1975), 153–184 .