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The first method uses multi-point Taylor expansions of\n \n \n $$ \\varvec{h(t,z)} $$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n ,<\/mml:mo>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n at certain appropriately selected base points that provides convergent expansions of the Lerch transcendent in terms of elementary functions of\n \n \n $$ \\varvec{z} $$<\/jats:tex-math>\n \n \n z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n uniformly valid in compact sets of the complex\n \n \n $$ \\varvec{z-} $$<\/jats:tex-math>\n \n \n z<\/mml:mi>\n -<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n plane. The second method expands\n \n \n $$ \\varvec{g(t,s,a)} $$<\/jats:tex-math>\n \n \n g<\/mml:mi>\n (<\/mml:mo>\n t<\/mml:mi>\n ,<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in a Taylor series at a selected point in\n \n \n $$ \\varvec{[0,1]} $$<\/jats:tex-math>\n \n \n [<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n giving a uniform convergent expansion of\n \n \n $$ \\varvec{\\Phi \\left( z,s,a\\right) } $$<\/jats:tex-math>\n \n \n \u03a6<\/mml:mi>\n \n z<\/mml:mi>\n ,<\/mml:mo>\n s<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n <\/mml:mfenced>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in terms of elementary functions of\n \n \n $$ \\varvec{z} $$<\/jats:tex-math>\n \n \n z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n valid in a large unbounded region of the complex plane. We provide explicit and\/or recursive algorithms for the computation of the coefficients of the expansions. Numerical experiments illustrate the accuracy of the new approximations.\n <\/jats:p>","DOI":"10.1007\/s11075-025-02113-w","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T01:30:07Z","timestamp":1748914207000},"page":"1613-1634","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New analytic representations of the Lerch transcendent"],"prefix":"10.1007","volume":"102","author":[{"given":"Jos\u00e9 L.","family":"L\u00f3pez","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ester","family":"P\u00e9rez Sinus\u00eda","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,6,3]]},"reference":[{"key":"2113_CR1","unstructured":"NIST Digital Library of Mathematical Functions. 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