{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T15:45:32Z","timestamp":1764603932121,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T00:00:00Z","timestamp":1674432000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LY21A010019","MOST 111-2115-M-017-002"],"award-info":[{"award-number":["LY21A010019","MOST 111-2115-M-017-002"]}]},{"name":"National Science and Technology Council of the Republic of China","award":["LY21A010019","MOST 111-2115-M-017-002"],"award-info":[{"award-number":["LY21A010019","MOST 111-2115-M-017-002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function xx, establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function xx and for related functions and integer sequences.<\/jats:p>","DOI":"10.3390\/sym15020323","type":"journal-article","created":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T03:48:08Z","timestamp":1674445688000},"page":"323","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Closed-Form Formulas for the nth Derivative of the Power-Exponential Function xx"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7173-0591","authenticated-orcid":false,"given":"Jian","family":"Cao","sequence":"first","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6239-2968","authenticated-orcid":false,"given":"Feng","family":"Qi","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China"},{"name":"Independent Researcher, Dallas, TX 75252-8024, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8996-2270","authenticated-orcid":false,"given":"Wei-Shih","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,23]]},"reference":[{"key":"ref_1","unstructured":"Comtet, L. 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Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley & Sons, Inc.","DOI":"10.1002\/9781118032572"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7494","DOI":"10.3934\/math.2021438","article-title":"Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions","volume":"6","author":"Guo","year":"2021","journal-title":"AIMS Math."},{"key":"ref_7","unstructured":"Charalambides, C.A. (2002). Enumerative Combinatorics, Chapman & Hall\/CRC."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"427","DOI":"10.2298\/AADM210401017G","article-title":"Maclaurin\u2019s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function","volume":"16","author":"Guo","year":"2022","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1515\/dema-2022-0157","article-title":"Taylor\u2019s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi","volume":"55","author":"Qi","year":"2022","journal-title":"Demonstr. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/323\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:13:56Z","timestamp":1760120036000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/323"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,23]]},"references-count":9,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020323"],"URL":"https:\/\/doi.org\/10.3390\/sym15020323","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,23]]}}}