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The sums are taken over finite and infinite domains defined in terms of the Hurwitz\u2013Lerch zeta function, which can be simplified to composite functions in special cases of integer values of the parameters involved. The results obtained include generalizations of finite and infinite products and sums of tangent, cotangent, hyperbolic tangent and hyperbolic cotangent functions, in certain cases raised to a complex number power.<\/jats:p>","DOI":"10.3390\/sym14081551","type":"journal-article","created":{"date-parts":[[2022,7,29]],"date-time":"2022-07-29T01:41:16Z","timestamp":1659058876000},"page":"1551","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extended Ohtsuka\u2013V\u0103lean Sums"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4230-9925","authenticated-orcid":false,"given":"Robert","family":"Reynolds","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7252-5004","authenticated-orcid":false,"given":"Allan","family":"Stauffer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"465","DOI":"10.4169\/amer.math.monthly.124.5.465","article-title":"Problems and Solutions","volume":"124","author":"Edgar","year":"2017","journal-title":"Am. 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[2nd ed.].","DOI":"10.1007\/978-0-387-48807-3"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"158","DOI":"10.29020\/nybg.ejpam.v15i1.4137","article-title":"A Note on the Infinite Sum of the Lerch function","volume":"15","author":"Reynolds","year":"2022","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.jmaa.2008.03.010","article-title":"A new approach to the representation of trigonometric and hyperbolic functions by infinite products","volume":"344","author":"Melnikov","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"117","DOI":"10.2140\/pjm.1956.6.117","article-title":"Note on the Lerch zeta function","volume":"6","author":"Oberhettinger","year":"1956","journal-title":"Pac. J. Math. Pac. J. 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