{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:42:39Z","timestamp":1760402559571,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T00:00:00Z","timestamp":1642118400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education","award":["NRF-2020R111A1A01052440"],"award-info":[{"award-number":["NRF-2020R111A1A01052440"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.<\/jats:p>","DOI":"10.3390\/axioms11010029","type":"journal-article","created":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T12:34:04Z","timestamp":1642163644000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Family of Generalized Legendre-Based Apostol-Type Polynomials"],"prefix":"10.3390","volume":"11","author":[{"given":"Talha","family":"Usman","sequence":"first","affiliation":[{"name":"Department of General Requirements, University of Technology and Applied Sciences, Sur 411, Oman"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nabiullah","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9285-0376","authenticated-orcid":false,"given":"Mohd","family":"Aman","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7240-7737","authenticated-orcid":false,"given":"Junesang","family":"Choi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Dongguk University, Gyeongju 38066, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,14]]},"reference":[{"key":"ref_1","first-page":"479","article-title":"Summation formulae of special functions and multivariable Hermite polynomials","volume":"119","author":"Dattoli","year":"2004","journal-title":"Nuovo Cimento Soc. 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