{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:42:06Z","timestamp":1760240526899,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,1]],"date-time":"2019-07-01T00:00:00Z","timestamp":1561939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2017R1E1A1A03070882"],"award-info":[{"award-number":["2017R1E1A1A03070882"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on     Z p    . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.<\/jats:p>","DOI":"10.3390\/sym11070847","type":"journal-article","created":{"date-parts":[[2019,7,1]],"date-time":"2019-07-01T10:54:47Z","timestamp":1561978487000},"page":"847","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials"],"prefix":"10.3390","volume":"11","author":[{"given":"Dmitry V.","family":"Dolgy","sequence":"first","affiliation":[{"name":"Hanrimwon, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9599-7015","authenticated-orcid":false,"given":"Dae San","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sogang University, Seoul 121-742, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jongkyum","family":"Kwon","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taekyun","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,1]]},"reference":[{"key":"ref_1","first-page":"288","article-title":"q-Volkenborn integration","volume":"9","author":"Kim","year":"2002","journal-title":"Russ. J. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Schikhof, W.H. (1984). Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511623844"},{"key":"ref_3","first-page":"51","article-title":"Degenerate Stirling, Bernoulli and Eulerian numbers","volume":"15","author":"Carlitz","year":"1979","journal-title":"Utilitas Math. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1186\/s13662-016-0764-z","article-title":"New results on higher-order Daehee and Bernoulli numbers and polynomials","volume":"2016","author":"Mustafa","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1007\/s11425-018-9338-5","article-title":"Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind","volume":"62","author":"Kim","year":"2019","journal-title":"Sci. China Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1186\/s13662-015-0610-8","article-title":"Extened q-Dedekind-type Daehee-Changhee sums associated with q-Euler polynomials","volume":"2015","author":"Araci","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1134\/S1061920809010063","article-title":"Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on \u2124p","volume":"16","author":"Kim","year":"2009","journal-title":"Russ. J. Math. Phys."},{"key":"ref_8","first-page":"15","article-title":"Sums of powers of consecutive q-integers","volume":"9","author":"Kim","year":"2004","journal-title":"Adv. Stud. Contemp. Math. (Kyungshang)"},{"key":"ref_9","first-page":"137","article-title":"A note on exploring the sums of powers of consecutive q-integers","volume":"11","author":"Kim","year":"2005","journal-title":"Adv. Stud. Contemp. Math. (Kyungshang)"},{"key":"ref_10","first-page":"117","article-title":"On the alternating sums of powers of consecutive integers","volume":"1","author":"Kim","year":"2005","journal-title":"J. Anal. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"611","DOI":"10.4134\/BKMS.2006.43.3.611","article-title":"On the alternating sums of powers of consecutive q-integers","volume":"43","author":"Rim","year":"2006","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_12","first-page":"69","article-title":"Exploring the q-analogues of the sums of powers of consecutive integers with Mathematica","volume":"18","author":"Ryoo","year":"2009","journal-title":"Adv. Stud. Contemp. Math. (Kyungshang)"},{"key":"ref_13","first-page":"81","article-title":"A note on the sums of powers of consecutive q-integers","volume":"1","author":"Simsek","year":"2016","journal-title":"J. Appl. Funct. Differ. Equ."},{"key":"ref_14","first-page":"319","article-title":"A note on degenerate Stirling polynomials of the second kind","volume":"20","author":"Kim","year":"2017","journal-title":"Proc. Jangjeon Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1007\/BF01900520","article-title":"A degenerate Staudt-Clausen theorem","volume":"7","author":"Carlitz","year":"1956","journal-title":"Arch. Math. (Basel)"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1216\/RMJ-2019-49-2-521","article-title":"Some identities involving special numbers and moments of random variables","volume":"49","author":"Kim","year":"2019","journal-title":"Rocky Mt. J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"69","DOI":"10.29020\/nybg.ejpam.v11i1.3153","article-title":"Application of probabilistic method on Daehee sequences","volume":"11","author":"Liu","year":"2018","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1134\/S1061920818010041","article-title":"Degenerate r-Stirling numbers and r-Bell polynomials","volume":"25","author":"Kim","year":"2018","journal-title":"Russ. J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1134\/S1061920817020091","article-title":"Degenerate Laplace transform and degenerate gamma function","volume":"24","author":"Kim","year":"2017","journal-title":"Russ. J. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/847\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:02:54Z","timestamp":1760187774000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/847"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,1]]},"references-count":19,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["sym11070847"],"URL":"https:\/\/doi.org\/10.3390\/sym11070847","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,7,1]]}}}