Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Seán Mark Stewart

doi:10.2478/tmmp-2020-0034

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Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

Tatra Mountains Mathematical Publications

Volume 77 (2020): Issue 1 (December 2020)

By: Seán Mark Stewart

Open Access

|Dec 2020

Abstract

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H₂_n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers H_n. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung $\sum_{n = 1}^{\infty} {(\frac{H_{n}}{n})}^{2} = \frac{17 π^{4}}{360}$ \sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \over n}} \right)}^2} = {{17{\pi ^4}} \over {360}}} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.

References

[1] BAILEY, D. H.—BORWEIN, J. M.—GIRGENSOHN, R.: Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17–30.
Search in Google Scholar Back to article
[2] BASTIEN, G.: Elementary methods for evaluating Jordan’s sums∑n⩾1(1+13+⋯⋯+12n-1)\sum {_{n \geqslant 1}\left( {1 + {1 \over 3} + \cdots \cdots + {1 \over {2n - 1}}} \right)} and ∑n⩾1(1+13+⋯+12n-1)1(2n-1)2α\sum {_{n \geqslant 1}\left( {1 + {1 \over 3} + \cdots + {1 \over {2n - 1}}} \right){1 \over {{{\left( {2n - 1} \right)}^{2\alpha }}}}}and analogous Euler’s type sums and for setting a σ-sum theorem, arXiv:1301.7662 (2013).
Search in Google Scholar Back to article
[3] BASU, A.—APOSTOL, T. M.: A new method for investigating Euler sums, Ramanujan J. 4 (2000), no. 4, 397–419.
Search in Google Scholar Back to article
[4] BORWEIN, D.—BORWEIN, J. M.: On an intriguing integral and some series related to ζ(4), Proc. Amer. Math. Soc. 123 (1995), no. 4, 1191–1198.
Search in Google Scholar Back to article
[5] BORWEIN, D.—BORWEIN, J. M.—GIRGENSOHN, R.: Explicit evaluation of Euler sums,Proc. Edinburgh Math.Soc. 38 (1995), no. 2, 277–294.
Search in Google Scholar Back to article
[6] BORWEIN, J. M.—BRADLEY, D. M.: Thirty-two Goldbach variations,Int.J.Number Theory 2 (2006), no. 1, 65–103.
Search in Google Scholar Back to article
[7] BOYADZHIEV, KH. N.: Power series with skew-harmonic numbers, dilogarithms, and double integrals, Tatra Mt. Math. Publ. 56 (2013), no. 1, 93–108.
Search in Google Scholar Back to article
[8] CHEN, H.: Evaluations of some variant Euler sums, J. Integer Seq. 9 (2006), no. 2, Article 06.2.3, 9 pp.
Search in Google Scholar Back to article
[9] CHEN, H.: Excursions in Classical Analysis. Pathways to Advanced Problem Solving and Undergraduate Research.In: Class. Resour. Mater. Mathematical Association of America, Washington, DC, 2010.
Search in Google Scholar Back to article
[10] CHEN, K.-W., EIE, M.: Explicit evaluations of extended Euler sums, J. Number Theory 117 (2006), no. 1, 31–52.
Search in Google Scholar Back to article
[11] CHOI, J.—SRIVASTAVA, H. M.: Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005), no. 1, 51–70.
Search in Google Scholar Back to article
[12] CHU, W.:, Hypergeometric series and the Riemann zeta function,Acta Arith. 82 (1997), no. 2, 103–118.
Search in Google Scholar Back to article
[13] DE DOELDER, P. J.: On some series containing ψ(x) – ψ(y) and (ψ(x) – ψ(y))²for certain values of x and y, J. Comput. Appl. Math. 37 (1991), no. 1–3, 125–141.
Search in Google Scholar Back to article
[14] FLAJOLET, P.—SALVY, B.: Euler sums and contour integral representations, Experment Math. 7 (1998), no. 1, 15–35.
Search in Google Scholar Back to article
[15] FREITAS, P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comp. 74 (2005), no. 251, 1425–1440.
Search in Google Scholar Back to article
[16] FURDUI, O.: Series involving products of two harmonic numbers, Math. Mag. 84 (2011), no. 5, 371–377.
Search in Google Scholar Back to article
[17] FURDUI, O.: Limits, Series, and Fractional Part Integrals.In: Problems in Mathematical Analysis. Problem Books in Mathematics. Springer, New York, 2013.10.1007/978-1-4614-6762-5
Search in Google Scholar Back to article
[18] FURDUI, O.—ŜINTĂMĂRIAN, A.: A new proof of the quadratic series of Au-Yeung, Gazeta Matematică Seria A 37 (2019), no. 1–2, 1–6.
Search in Google Scholar Back to article
[19] GRADSHTEYN, I. S.—RYZHIK, I. M.: Table of Integrals, Series and Products.6th ed. (Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger.) Academic Press, Inc., San Diego, CA, 2000.
Search in Google Scholar Back to article
[20] HANSEN, E. R.: A Table of Series and Products. Prentice-Hall, Series in Automatic Computation. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975.
Search in Google Scholar Back to article
[21] JORDAN, P. F.: Infinite sums of psi functions, Bull. Amer. Math. Soc. 79 (1973), no. 4, 681–683.
Search in Google Scholar Back to article
[22] KNESER, M.: A summation problem. Solution to Problem 4305, Amer. Math. Monthly 57 (1950), no. 4, 267–268.
Search in Google Scholar Back to article
[23] KOUBA, O.: The sum of certain series related to harmonic numbers, Octogon Math. Mag. 19 (2011), no. 1, 3–18.
Search in Google Scholar Back to article
[24] LEWIN, L.: Polylogarithms and Associated Functions. North Holland, New York, 1981.
Search in Google Scholar Back to article
[25] MELVILLE, J.: A simple series representation for Apéry’s constant, Math. Gaz. 97 (2006), no. 540, 455–460.
Search in Google Scholar Back to article
[26] MEZŐ, I.: A q-Raabe formula and an integral of the fourth Jacobi theta function, J. Number Theory 133 (2013), no. 2, 693–704.
Search in Google Scholar Back to article
[27] MEZŐ, I.: Nonlinear Euler sums,Pacific J.Math. 272 (2014), no. 1, 201–226.
Search in Google Scholar Back to article
[28] OLDHAM, K.—MYLAND, J.—SPANIER, J.: An Atlas of Functions. 2nd ed. Springer, New York, 2009.10.1007/978-0-387-48807-3
Search in Google Scholar Back to article
[29] OLVER, F. W. J.—LOZIER, D. W.—BOISVERT, R. F.—CLARK, C. W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, 2010.
Search in Google Scholar Back to article
[30] ONG, Y. L.—EIE, M.: On recurrence relations for the extensions of Euler sums,Rocky Mountain J. Math. 38 (2008), no. 1, 225–251.
Search in Google Scholar Back to article
[31] RĂDULESCU, T.-L.—RĂDULESCA, V. D.—ANDREESCU, T.: Problems in Real Analysis: Advanced Calculus on the Real Line. Springer, New York, 2009.
Search in Google Scholar Back to article
[32] SANDHAM, H. F.: Problem 4305,Amer. Math. Monthly 55 (1948), no. 7, 431.
Search in Google Scholar Back to article
[33] SHATHER, A.: Mathematics Stack Exchange: math.stackexchange.com/q/3361723 (2019).
Search in Google Scholar Back to article
[34] SHEN, L.-C.: Remarks on some integrals and series involving the Stirling numbers and ζ(n), Trans. Amer. Math. Soc. 347 (1995), no. 4, 1391–1399.
Search in Google Scholar Back to article
[35] SITARAMACHANDRARAO, R.: A formula of S. Ramanujan, J. Number Theory 25 (1987), no. 1, 1–19.
Search in Google Scholar Back to article
[36] SOFO, A.: Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), 144–159.10.1016/j.jnt.2015.02.013
Search in Google Scholar Back to article
[37] SOFO, A.: General order Euler sums with rational argument, Integral Transforms Spec. Funct. 30 (2019), no. 12, 1–14.
Search in Google Scholar Back to article
[38] SOFO, A.—NIMBRAN, A. S.: Euler sums and integral connections, Mathematics 7 (2019), no. 9, Article 833.
Search in Google Scholar Back to article
[39] VĂLEAN, C. I.—FURDUI, O.: Reviving the quadratic series of Au–Yeung, J. Class. Anal. 6 (2015), no. 2, 113–118.
Search in Google Scholar Back to article
[40] VĂLEAN, C. I.: (Almost) Impossible Integrals, Sums, and Series. (With a foreword by Paul J. Nahin.) In: Problem Books in Mathematics. Springer, Cham, 2019.
Search in Google Scholar Back to article
[41] XU, C.—CHENG, J.: Some results on Euler sums, Funct. Approx. Comment. Math. 54 (2016), no. 1, 25–37.
Search in Google Scholar Back to article
[42] XU, C.—YANG, Y. Y.—ZHANG, J.: Explicit evaluation of quadratic Euler sums,Int.J. Number Theory 13 (2017), no. 3, 655–672.
Search in Google Scholar Back to article
[43] XU, C.: Evaluations of Euler-type sums of weight ⩽ 5, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 1, 847–877.
Search in Google Scholar Back to article
[44] ZHAO, M. H.: On specific log integrals, polylog integrals and alternating Euler sums, (2020), arXiv:1911.12155v6.
Search in Google Scholar Back to article
[45] ZHENG, D.-Y.: Further summation formulae related to generalized harmonic numbers, J. Math. Anal. Appl. 335 (2007), no. 1, 692–706.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2020-0034 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 73 - 98

Submitted on: Mar 15, 2020

Published on: Dec 31, 2020

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Euler sums,

Harmonic number,

Generating function,

Polylogarithm function,

Riemann zeta function

Related subjects:

Mathematics,

General mathematics

© 2020 Seán Mark Stewart, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 77 (2020): Issue 1 (December 2020)