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Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given.<\/jats:p>","DOI":"10.3390\/sym17071126","type":"journal-article","created":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T08:04:41Z","timestamp":1752566681000},"page":"1126","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Formulae for Generalization of Touchard Polynomials with Their Generating Functions"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8051-2879","authenticated-orcid":false,"given":"Ayse","family":"Yilmaz Ceylan","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Akdeniz, 07070 Antalya, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0611-7141","authenticated-orcid":false,"given":"Yilmaz","family":"Simsek","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Akdeniz, 07070 Antalya, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"258","DOI":"10.2307\/1968431","article-title":"Exponential Polynomials","volume":"35","author":"Bell","year":"1934","journal-title":"Ann. 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