{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:08:00Z","timestamp":1760242080865,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,11,29]],"date-time":"2018-11-29T00:00:00Z","timestamp":1543449600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character \u03c7 and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities.<\/jats:p>","DOI":"10.3390\/sym10120675","type":"journal-article","created":{"date-parts":[[2018,11,29]],"date-time":"2018-11-29T11:47:53Z","timestamp":1543492073000},"page":"675","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetric Identities of Hermite-Bernoulli Polynomials and Hermite-Bernoulli Numbers Attached to a Dirichlet Character \u03c7"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3950-6864","authenticated-orcid":false,"given":"Serkan","family":"Araci","sequence":"first","affiliation":[{"name":"Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Waseem Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Integral University, Lucknow-226026, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5769-4320","authenticated-orcid":false,"given":"Kottakkaran Sooppy","family":"Nisar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Science-Wadi Aldawaser, Prince Sattam bin Abdulaziz University, 11991 Riyadh Region, Kingdom of Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,29]]},"reference":[{"key":"ref_1","first-page":"288","article-title":"q-Volkenborn integration","volume":"9","author":"Kim","year":"2002","journal-title":"Russ. 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