Another Euler Constant: The Value of an Euler Integral

[1] ALLOUCHE, J.-P.: A note on products involving ζ(3) and Catalan’s constant, Ramanujan J. 37 (2015), 79–88. Search in Google Scholar

[2] ALLOUCHE, J.-P.: Hölder and Kurokawa meet Borwein-Dykshoorn and Adamchik, J. Ramanujan Math. Soc. 38 (2023), 265–273. Search in Google Scholar

[3] ALMKVIST, G.: Glaisher’s formulas for 1/π² and some generalizations. (With an appendix by MEURMAN, A.). In: Advances in Combinatorics (I. S. Kotsireas, E. V. Zima, Eds.), Springer, Heidelberg, 2013, pp. 1–21. Search in Google Scholar

[4] BERNDT, B. C.: Ramanujan’s Notebooks. Part I. Springer-Verlag, New York, 1985. Search in Google Scholar

[5] BOROS, G.—MOLL, V.: Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge, 2004. Search in Google Scholar

[6] BORWEIN, J. M.—CHAMBERLAND, M.: Integer powers of arcsin, Int. J. Math. Math. Sci. (2007), Art. ID 19381. Search in Google Scholar

[7] BORWEIN, J. M.—STRAUB, A.—WAN, J.: Three-step and four-step random walk integrals, Exp. Math. 22 (2013), 1–14. Search in Google Scholar

[8] BOYD, D. W.: Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24 (1981), 453–469. Search in Google Scholar

[9] CAMPBELL, J. M.: On a higher-order version of a formula due to Ramanujan, Integral Transforms Spec. Funct. 35 (2024), 260–269. Search in Google Scholar

[10] CANTARINI, M.: A note on Clebsch-Gordan integral, Fourier-Legendre expansions and closed form for hypergeometric series, Ramanujan J. 59 (2022), 549–557. Search in Google Scholar

[11] CANTARINI, M.—D’AURIZIO, J.: On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums, Boll. Unione Mat. Ital. 12 (2019), 623–656. Search in Google Scholar

[12] CHEN H.: A power series expansion and its applications, Internat. J. Math. Ed. Sci. Tech. 37 (2006), 362–368. Search in Google Scholar

[13] CHEN, H.: Evaluations of some variant Euler sums, J. Integer Seq. 9 (2006), Art. 06.2.3. Search in Google Scholar

[14] CHEN, H.: Interesting Ramanujan-like series associated with powers of central binomial coefficients, J. Integer Seq. 25 (2022), no. 1, Art. 22.1.8. Search in Google Scholar

[15] CHEN, M.-P.—SRIVASTAVA, H. M.: Some families of series representations for the Riemann ζ(3), Results Math. 33 (1998), no. 3–4, 179–197. Search in Google Scholar

[16] CHOI, J.—SRIVASTAVA, H. M.: Certain classes of series involving the zeta function, J. Math. Anal. Appl. 231 (1999), no. 1, 91–117. Search in Google Scholar

[17] CHOI, J.—SRIVASTAVA, H. M.: Certain classes of series associated with the Zeta function and multiple gamma functions, J. Comput. Appl. Math. 118 (2000), 87–109. Search in Google Scholar

[18] CHU, W.: π-formulas implied by Dougall’s summation theorem for ₅F₄-series, Ramanujan J. 26 (2011), 251–255. Search in Google Scholar

[19] COPPO, M.-A.—CANDELPERGHER, B.: Inverse binomial series and values of Arakawa-Kaneko zeta functions, J. Number Theory 150 (2015), 98–119. Search in Google Scholar

[20] DĄBROWSKI, A.: A note on values of the Riemann zeta function at positive odd integers, Nieuw Arch. Wisk. 14 (1996), 199–207. Search in Google Scholar

[21] DOUGALL, J.: On Vandermonde’s theorem, and some more general expansions, Proc. Edinb. Math. Soc. 25 (1907), 114–132. Search in Google Scholar

[22] EULER, L.: Exercitationes analyticae, Novi commentarii academiae scientiarum Petropolitanae 17 (1772), 173–204; available at: https://scholarlycommons.pacific.edu/euler-works/432/. Also see: Opera Omnia, 1–15, pp. 131–167. Search in Google Scholar

[23] FINCH, S. R.: Mathematical Constants. Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Cambridge, 2003. Search in Google Scholar

[24] GLAISHER, J. W. L.: On the series for 1/π and 1/π², Quart. J. 37 (1905), 173–198. Search in Google Scholar

[25] GUTNIK, L. A.: On the irrationality of some quantities containing ζ(3) (English. Russian original) Zbl 0657.10036 Transl., Ser. 2, Am. Math. Soc. 140, 45–55 (1988); (translation from Acta Arith. 42, 255–264 (1983) Search in Google Scholar

[26] HU, S.—KIM, M.-S.: Euler’s integral, multiple cosine function and zeta values, Preprint (2023), available at: https://arxiv.org/abs/2201.01124; to appear in Forum Math. https://doi.org/10.1515/forum-2023-0426. Search in Google Scholar

[27] KOYAMA, S.-Y.—KUROKAWA, N.: Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133 (2005), 1257–1265. Search in Google Scholar

[28] KUROKAWA, N.: Multiple sine functions and Selberg zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 67 (1991), 61–64. Search in Google Scholar

[29] KUROKAWA, N.: Multiple zeta functions: an example. In: Zeta Functions in Geometry (Tokyo, 1990), Adv. Stud. Pure Math. Vol. 21, Kinokuniya, Tokyo, 1992, pp. 219–226. Search in Google Scholar

[30] KUROKAWA, N.—KOYAMA, S.-Y.: Multiple sine functions, Forum Math. 15 (2003), 839–876. Search in Google Scholar

[31] KUROKAWA, N.—OCHIAI, H.: Mahler measures via the crystalization, Comment. Math. Univ. St. Pauli 54 (2005), 121–137. Search in Google Scholar

[32] LAGARIAS, J. C.: Euler’s constant: Euler’s work and modern developments, Bull. Amer. Math. Soc. 50 (2013), 527–628. Search in Google Scholar

[33] LEVRIE, P.: Using Fourier-Legendre expansions to derive series for 1/π and 1/π², Ramanujan J. 22 (2010), 221–230. Search in Google Scholar

[34] LEVRIE, P.—CAMPBELL, J.: Series acceleration formulas obtained from experimentally discovered hypergeometric recursions, Discrete Math. Theor. Comput. Sci. 24 (2022), Art. no. 12. Search in Google Scholar

[35] LIU, Z.-G.: A summation formula and Ramanujan type series, J. Math. Anal. Appl. 389 (2012), 1059–1065. Search in Google Scholar

[36] MUZAFFAR, H.: Some interesting series arising from the power series expansion of (sin⁻¹ x)^q, Int. J. Math. Math. Sci. (2005), 2329–2336. Search in Google Scholar

[37] NASH, C.— O’CONNOR, D. J.: Ray-Singer torsion, topological field theories and the Riemann zeta function at s = 3. In: Low-dimensional Topology and Quantum Field Theory (Cambridge, 1992), NATO Adv. Sci. Inst. Ser. B: Phys. 315, Plenum, New York, 1993, pp. 279–288. Also available at: https://arxiv.org/pdf/hep-th/9210005.pdf. Search in Google Scholar

[38] NASH, C.—O’CONNOR, D.: Determinants of Laplacians, the Ray-Singer torsion on lens spaces and the Riemann zeta function, J. Math. Phys. 36 (1995), no. 3, 1462–1505. Erratum: J. Math. Phys. 36 (1995), no. 8, 4549. Search in Google Scholar

[39] NIMBRAN, A. S.: Deriving Forsyth-Glaisher type series for 1/π and Catalan’s constant by an elementary method, Math. Student 84 (2015), 69–86. Search in Google Scholar

[40] NIMBRAN, A. S.—LEVRIE, P.—SOFO, A.: Harmonic-binomial Euler-like sums via expansions of (arcsin x)^p, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., RACSAM 116 (2022), Art. no. 23. Search in Google Scholar

[41] PAPANIKOLAS, M. A.—ROGERS, M. D.—SAMART, D.: The Mahler measure of a Calabi-Yau threefold and special L-values, Math. Z. 276 (2014), 1151–1163. Search in Google Scholar

[42] ROGERS, M.—ZUDILIN, W.: On the Mahler measure of 1+ X +1/X + Y +1/Y, Int. Math. Res. Not. IMRN 9 (2014), 2305–2326. Search in Google Scholar

[43] SHERMAN, T.: Summation of Glaisher - and Apéry - like series, Preprint (2000), https://web.archive.org/web/20221203114652. https://math.arizona.edu/~rta/001/sherman.travis/series.pdf. Search in Google Scholar

[44] SRIVASTAVA, H. M.—CHOI, J.: Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdamm, 2012. Search in Google Scholar

[45] SRIVASTAVA, H. M.—GLASSER, M. L.—ADAMCHIK, V. S.: Some definite integrals associated with the Riemann zeta function, Z. Anal. Anwendungen 19 (2000), no. 3, 831–846. Search in Google Scholar

[46] WANG, W.—XU, C.: Alternating multiple zeta values, and explicit formulas of some Euler-Apéry-type series, European J. Combin. 93 (2021), Art. no. 10328. Search in Google Scholar

[47] XU, C.—WANG, W.: Two variants of Euler sums, Monatsh. Math. 199 (2022), 431–454. Search in Google Scholar

[48] XU, C.—WANG, W.: Dirichlet type extensions of Euler sums, C. R. Math. Acad. Sci. Paris 361 (2023), 979–1010. Search in Google Scholar

[49] YUE, Z. N.—WILLIAMS, K. S.: Some series representations of ζ(2n +1), Rocky Mountain J. Math. 23 (1993), 1581–1592. Search in Google Scholar