{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T05:13:09Z","timestamp":1775365989798,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,1]],"date-time":"2018-10-01T00:00:00Z","timestamp":1538352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)","award":["2017R1A2B4006092"],"award-info":[{"award-number":["2017R1A2B4006092"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The q-Bernoulli numbers and polynomials can be given by Witt\u2019s type formulas as p-adic invariant integrals on     Z p    . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on     Z p     of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.<\/jats:p>","DOI":"10.3390\/sym10100451","type":"journal-article","created":{"date-parts":[[2018,10,2]],"date-time":"2018-10-02T08:23:50Z","timestamp":1538468630000},"page":"451","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9599-7015","authenticated-orcid":false,"given":"Dae","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sogang University, Seoul 121-742, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taekyun","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China"},{"name":"Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheon","family":"Ryoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 306-791, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0452-785X","authenticated-orcid":false,"given":"Yonghong","family":"Yao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China"},{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1134\/S1061920811020014","article-title":"Identities involving values of Bernstein q-Bernoulli, and q-Euler polynomials","volume":"18","author":"Bayad","year":"2011","journal-title":"Russ. 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