{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:31:04Z","timestamp":1772253064795,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,15]],"date-time":"2020-10-15T00:00:00Z","timestamp":1602720000000},"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>Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.<\/jats:p>","DOI":"10.3390\/sym12101691","type":"journal-article","created":{"date-parts":[[2020,10,17]],"date-time":"2020-10-17T07:23:22Z","timestamp":1602919402000},"page":"1691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mehmet","family":"Acikgoz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, University of Gaziantep, TR-27310 Gaziantep, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5717-1199","authenticated-orcid":false,"given":"Ugur","family":"Duran","sequence":"additional","affiliation":[{"name":"Department of the Basic Concepts of Engineering, Faculty of Engineering and Natural Sciences, Iskenderun Technical University, TR-31200 Hatay, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,15]]},"reference":[{"key":"ref_1","first-page":"51","article-title":"Degenerate Stirling, Bernoulli and Eulerian numbers","volume":"15","author":"Carlitz","year":"1979","journal-title":"Util. Math."},{"key":"ref_2","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."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/S0893-9659(03)80058-7","article-title":"A note on the Bernoulli and Euler polynomials","volume":"16","author":"Cheon","year":"2003","journal-title":"Appl. Math. Lett."},{"key":"ref_4","unstructured":"Dutta, H., and Peters, J. (2020). Multi poly-Bernoulli and multi poly-Euler polynomials. Applied Mathematical Analysis: Theory, Methods, and Applications, Springer. Studies in Systems, Decision and Control."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1186\/s13662-020-02889-2","article-title":"Construction of the type 2 poly-Frobenius-Genocchi polynomials with their certain applications","volume":"2020","author":"Duran","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Duran, U., and Acikgoz, M. (2020). On degenerate truncated special polynomials. Mathematics, 8.","DOI":"10.3390\/math8010144"},{"key":"ref_7","first-page":"1","article-title":"A new approach to the Poisson distribution: Degenerate Poisson distribution","volume":"11","author":"Duran","year":"2020","journal-title":"J. Inequal. Spec. Funct."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Duran, U., and Sadjang, P.N. (2019). On Gould-Hopper-based fully degenerate poly-Bernoulli polynomials with a q-parameter. Mathematics, 7.","DOI":"10.3390\/math7020121"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1017\/S2040618500034961","article-title":"On polylogarithms","volume":"6","author":"Eastham","year":"1964","journal-title":"Proc. Glasgow Math. Assoc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s40590-015-0061-y","article-title":"More properties on multi-poly-Euler polynomials","volume":"21","author":"Jolany","year":"2015","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"7","DOI":"10.7169\/facm\/2014.51.1.1","article-title":"Poly-Euler polynomials and Arakawa-Kaneko type zeta functions","volume":"51","author":"Hamahata","year":"2014","journal-title":"Funct. Approx. Comment. Math."},{"key":"ref_12","first-page":"97","article-title":"Multiple poly-Bernoulli polynomials of the second kind associated with Hermite polynomials","volume":"58","author":"Khan","year":"2017","journal-title":"Fasc. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1134\/S1061920819010047","article-title":"A note on polyexponential and unipoly functions","volume":"26","author":"Kim","year":"2019","journal-title":"Russ. J. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kim, T., Kim, D.S., Kwon, J., and Lee, H. (2020). Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials. Adv. Differ. Equ., 168.","DOI":"10.1186\/s13662-020-02636-7"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1134\/S1061920820020090","article-title":"A note on a new type of degenerate Bernoulli numbers","volume":"27","author":"Kim","year":"2020","journal-title":"Russ. J. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kim, T., Kim, D.S., Kwon, J., and Kim, H.Y. (2020). A note on degenerate Genocchi and poly-Genocchi numbers and polynomials. J. Ineq. Appl., 110.","DOI":"10.1186\/s13660-020-02378-w"},{"key":"ref_17","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_18","doi-asserted-by":"crossref","first-page":"124017","DOI":"10.1016\/j.jmaa.2020.124017","article-title":"Degenerate polyexponential functions and degenerate Bell polynomials","volume":"487","author":"Kim","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kim, T., and Kim, D.S. (2020). A note on degenerate multi-poly-Bernoulli numbers and polynomials. arXiv.","DOI":"10.1186\/s13662-020-02901-9"},{"key":"ref_20","first-page":"447","article-title":"A note on degenerate multi-poly-Genocchi polynomials","volume":"30","author":"Kim","year":"2020","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kim, T., Khan, W.A., Sharma, S.K., and Ghayasuddin, M. (2020). A note on parametric kinds of the degenerate poly-Bernoulli and poly-Genocchi polynomials. Symmetry, 12.","DOI":"10.3390\/sym12040614"},{"key":"ref_22","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."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kim, T., Kim, D., Kim, H.-Y., Lee, H., and Jang, L.-C. (2020). Degenerate poly-Bernoulli polynomials arising from degenerate polylogarithm. Adv. Differ. Equ., 444.","DOI":"10.1186\/s13662-020-02901-9"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1134\/S1061920820030061","article-title":"Note on the degenerate gamma function","volume":"27","author":"Kim","year":"2020","journal-title":"Russ. J. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lee, D.S., Kim, H.-Y., and Jang, L.-C. (2020). Type 2 degenerate poly-Euler polynomials. Symmetry, 12.","DOI":"10.3390\/sym12061011"},{"key":"ref_26","first-page":"1","article-title":"Multiple poly-Bernoulli polynomials of the second kind","volume":"25","author":"Qi","year":"2015","journal-title":"Adv. Stud. Contemp. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1691\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:21:32Z","timestamp":1760178092000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1691"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,15]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101691"],"URL":"https:\/\/doi.org\/10.3390\/sym12101691","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202008.0706.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,15]]}}}