{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:09:04Z","timestamp":1760058544839,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T00:00:00Z","timestamp":1744329600000},"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 examining Fourier\u2013Legendre series pairs F(y) and G(y), fifteen classes of symmetric and asymmetric bivariate series will systematically be investigated. A compendium of 360 remarkable summation formulae will be established. They may serve as a complementary work to the paper by Chu and Campbell (The Ramanujan Journal (2023)).<\/jats:p>","DOI":"10.3390\/axioms14040287","type":"journal-article","created":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T10:05:51Z","timestamp":1744365951000},"page":"287","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Symmetric and Asymmetric Double Series via Expansions over Legendre Polynomials"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2012-1235","authenticated-orcid":false,"given":"Mei","family":"Bai","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":[[2025,4,11]]},"reference":[{"key":"ref_1","unstructured":"Bailey, W.N. (1935). Generalized Hypergeometric Series, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1007\/s40574-019-00198-5","article-title":"On the interplay between hypergeometric series, Fourier\u2013Legendre expansions and Euler sums","volume":"12","author":"Cantarini","year":"2019","journal-title":"Boll. Unione Mat. Ital."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.apnum.2024.03.014","article-title":"Legendre expansions of products of functions with applications to nonlinear partial differential equations","volume":"201","author":"Djellouli","year":"2024","journal-title":"Appl. Numer. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4064\/ap200709-4-1","article-title":"New families of double hypergeometric series for constants involving 1\/\u03c02","volume":"126","author":"Campbell","year":"2021","journal-title":"Ann. Polon. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s11139-022-00663-4","article-title":"Expansinos over Legendre polynomials and infinite double series identitie","volume":"60","author":"Chu","year":"2023","journal-title":"Ramanujan J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"817","DOI":"10.4171\/zaa\/1110","article-title":"A certain series associated with Catalan\u2019s constant","volume":"21","author":"Adamchik","year":"2002","journal-title":"Z. Anal. Anwendungen"},{"key":"ref_7","unstructured":"Marichev, O., Sondow, J., and Weisstein, E.M. (2025, February 25). Catalan\u2019s Constant: From MathWorld\u2014A Wolfram Web Resource. Available online: https:\/\/mathworld.wolfram.com\/CatalansConstant.html."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1017\/S2040618500033104","article-title":"Elliptic integrals in terms of Legendre polynomials","volume":"2","year":"1954","journal-title":"Proc. Glasgow Math. Assoc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1063\/1.1665035","article-title":"Legendre polynomial expansions of hypergeometric functions with applications","volume":"11","author":"Holdeman","year":"1970","journal-title":"J. Math. Phys."},{"key":"ref_10","unstructured":"Rainville, E.D. (1960). Special Functions, The Macmillan Company."},{"key":"ref_11","unstructured":"Weisstein, E.W. (2025, February 25). Dilogarithm: From MathWorld\u2014A Wolfram Web Resource. Available online: https:\/\/mathworld.wolfram.com\/Dilogarithm.html."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/287\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:12:52Z","timestamp":1760029972000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/287"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,11]]},"references-count":11,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040287"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040287","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,11]]}}}