{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:27:00Z","timestamp":1760059620571,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T00:00:00Z","timestamp":1750809600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>By means of the coefficient extraction method, we examine a transformation of a classical hypergeometric series. Three classes of infinite series (of convergence rate \u201c1\/4\u201d) with harmonic numbers in numerators and cubic central binomial coefficients in denominators are expressed in terms of odd Euler sums. Several new closed formulae are established.<\/jats:p>","DOI":"10.3390\/axioms14070495","type":"journal-article","created":{"date-parts":[[2025,6,26]],"date-time":"2025-06-26T05:53:13Z","timestamp":1750917193000},"page":"495","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Odd Euler Sums and Harmonic Series with Cubic Central Binomial Coefficients in Denominators"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2464-5842","authenticated-orcid":false,"given":"Chunli","family":"Li","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8425-212X","authenticated-orcid":false,"given":"Wenchang","family":"Chu","sequence":"additional","affiliation":[{"name":"Independent Researcher, Via Dalmazio Birago 9\/E, 73100 Lecce, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Adegoke, K., Frontczak, R., and Goy, T. (2022). On a family of infinite series with reciprocal Catalan numbers. Axioms, 11.","DOI":"10.3390\/axioms11040165"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s13398-021-01025-3","article-title":"Parametric binomial sums involving harmonic numbers","volume":"115","author":"Batir","year":"2021","journal-title":"RACSAM"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Komatsu, T., and Sury, B. (2025). Polynomial identities for binomial sums of harmonic numbers of higher order. Mathematics, 13.","DOI":"10.3390\/math13020321"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4611","DOI":"10.3934\/era.2023236","article-title":"Infinite series about harmonic numbers inspired by Ramanujan\u2013like formulae","volume":"31","author":"Li","year":"2023","journal-title":"Electron. Res. Arch"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1080\/10236198.2024.2388746","article-title":"Hypergeometric series and generalized harmonic numbers","volume":"31","author":"Chen","year":"2025","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Li, C.L., and Chu, W. (2023). Series of convergence rate \u22121\/4 containing harmonic numbers. Axioms, 12.","DOI":"10.3390\/axioms12060513"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2223","DOI":"10.1090\/S0025-5718-2011-02474-9","article-title":"Dougall\u2019s bilateral 2H2-series and Ramanujan\u2013like \u03c0-formulae","volume":"80","author":"Chu","year":"2011","journal-title":"Math. Comp."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2513","DOI":"10.1216\/RMJ-2019-49-8-2513","article-title":"New series involving harmonic numbers and squared central binomial coefficients","volume":"49","author":"Campbell","year":"2019","journal-title":"Rocky Mt. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1080\/10652469.2017.1318874","article-title":"An integral transform related to series involving alternating harmonic numbers","volume":"28","author":"Campbell","year":"2017","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s11139-019-00140-5","article-title":"Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients","volume":"52","author":"Wang","year":"2020","journal-title":"Ramanujan J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"529","DOI":"10.4310\/CNTP.2019.v13.n3.a2","article-title":"An odd variant of multiple zeta values","volume":"13","author":"Hoffman","year":"2019","journal-title":"Commun. Number Theory Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1007\/s41478-022-00472-4","article-title":"On explicit evaluation of certain linear alternating Euler sums and double t-values","volume":"31","author":"Chavan","year":"2022","journal-title":"J. Anal."},{"key":"ref_13","unstructured":"V\u01celean, C.I. (2025, May 25). A New Powerful Strategy of Calculating a Class of Alternating Euler Sums. ResearchGate. Available online: https:\/\/www.researchgate.net\/publication\/333999069."},{"key":"ref_14","first-page":"11","article-title":"Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers","volume":"15","author":"Boyadzhiev","year":"2012","journal-title":"J. Integer Seq."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"215","DOI":"10.2478\/s12175-011-0006-5","article-title":"Binomial sums involving harmonic numbers","volume":"61","year":"2011","journal-title":"Math. Slovaca."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1017\/mag.2017.4","article-title":"Integrals evaluated in terms of Catalan\u2019s constant","volume":"101","author":"Jameron","year":"2017","journal-title":"Math. Gaz."},{"key":"ref_17","unstructured":"Lewin, L. (1981). Polylogarithms and Associated Functions, North-Holland."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"123","DOI":"10.7153\/jca-08-11","article-title":"Harmonic series with polygamma functions","volume":"8","author":"Furdui","year":"2016","journal-title":"J. Class. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Comtet, L. (1974). Advanced Combinatorics, Springer.","DOI":"10.1007\/978-94-010-2196-8"},{"key":"ref_20","unstructured":"Olaikhan, A.S. (2023). An Introduction to the Harmonic Series and Logarithmic Integrals, Ali Shadhar Olaikhan, Amazon Digital Services LLC KDP Print US. [2nd ed.]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/495\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:58:36Z","timestamp":1760032716000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/495"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,25]]},"references-count":20,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070495"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070495","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,6,25]]}}}