{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T08:22:20Z","timestamp":1768292540695,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T00:00:00Z","timestamp":1590969600000},"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>Harmonic numbers appear, for example, in many expressions involving Riemann zeta functions. Here, among other things, we introduce and study discrete versions of those numbers, namely the discrete harmonic numbers. The aim of this paper is twofold. The first is to find several relations between the Type 2 higher-order degenerate Euler polynomials and the Type 2 high-order Changhee polynomials in connection with the degenerate Stirling numbers of both kinds and Jindalrae\u2013Stirling numbers of both kinds. The second is to define the discrete harmonic numbers and some related polynomials and numbers, and to derive their explicit expressions and an identity.<\/jats:p>","DOI":"10.3390\/sym12060905","type":"journal-article","created":{"date-parts":[[2020,6,3]],"date-time":"2020-06-03T04:12:09Z","timestamp":1591157529000},"page":"905","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Some Relations of Two Type 2 Polynomials and Discrete Harmonic Numbers and Polynomials"],"prefix":"10.3390","volume":"12","author":[{"given":"Taekyun","family":"Kim","sequence":"first","affiliation":[{"name":"School of Sciences, Xian Technological University, Xi\u2019an 710021, China"},{"name":"Department of Mathematics, 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"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,1]]},"reference":[{"key":"ref_1","first-page":"51","article-title":"Degenerate Stirling, Bernoulli and Eulerian numbers","volume":"15","author":"Carlitz","year":"1979","journal-title":"Utilitas Math."},{"key":"ref_2","first-page":"613","article-title":"Some identities of the Type 2 degenerate Bernoulli and Euler numbers","volume":"29","author":"Chung","year":"2019","journal-title":"Adv. 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