{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T19:31:52Z","timestamp":1769023912233,"version":"3.49.0"},"reference-count":38,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,2,20]],"date-time":"2012-02-20T00:00:00Z","timestamp":1329696000000},"content-version":"vor","delay-in-days":50,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Zhejiang Province Natural Science Foundation","award":["Y6110310"],"award-info":[{"award-number":["Y6110310"]}]},{"name":"Zhejiang Province Natural Science Foundation","award":["Y6100021"],"award-info":[{"award-number":["Y6100021"]}]},{"name":"Zhejiang Province Natural Science Foundation","award":["Y6110310"],"award-info":[{"award-number":["Y6110310"]}]},{"name":"Zhejiang Province Natural Science Foundation","award":["Y6100021"],"award-info":[{"award-number":["Y6100021"]}]},{"DOI":"10.13039\/100007834","name":"Ningbo Natural Science Foundation","doi-asserted-by":"crossref","award":["Y6110310"],"award-info":[{"award-number":["Y6110310"]}],"id":[{"id":"10.13039\/100007834","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100007834","name":"Ningbo Natural Science Foundation","doi-asserted-by":"crossref","award":["Y6100021"],"award-info":[{"award-number":["Y6100021"]}],"id":[{"id":"10.13039\/100007834","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We employ the generalized factorials to define a Stirling\u2010type pair {<jats:italic>s<\/jats:italic>(<jats:italic>n<\/jats:italic>,<jats:italic>k<\/jats:italic>;<jats:bold>\u03b1<\/jats:bold>,<jats:bold>\u03b2<\/jats:bold>,<jats:italic>r<\/jats:italic>),<jats:italic>S<\/jats:italic>(<jats:italic>n<\/jats:italic>,<jats:italic>k<\/jats:italic>;<jats:bold>\u03b1<\/jats:bold>,<jats:bold>\u03b2<\/jats:bold>,<jats:italic>r<\/jats:italic>)} which unifies various Stirling\u2010type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well\u2010known Dobinski\u2032s formula are further investigated.<\/jats:p>","DOI":"10.1155\/2012\/351935","type":"journal-article","created":{"date-parts":[[2012,2,20]],"date-time":"2012-02-20T21:01:38Z","timestamp":1329771698000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Newton Interpolation Approach to Generalized Stirling Numbers"],"prefix":"10.1155","volume":"2012","author":[{"given":"Aimin","family":"Xu","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,2,20]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1006\/aama.1998.0586"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177732430"},{"key":"e_1_2_6_3_2","first-page":"51","article-title":"Degenerate Stirling, Bernoulli and Eulerian numbers","volume":"15","author":"Carlitz L.","year":"1979","journal-title":"Utilitas Mathematica"},{"key":"e_1_2_6_4_2","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1080\/00150517.1980.12430154","article-title":"Weighted Stirling numbers of the first and second kind\u2014II","volume":"18","author":"Carlitz L.","year":"1980","journal-title":"The Fibonacci Quarterly"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(85)90155-4"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90089-4"},{"key":"e_1_2_6_7_2","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-62-02907-1"},{"key":"e_1_2_6_8_2","first-page":"143","volume-title":"Combinatorial-Algebraic Methods in Applied Mathematics","author":"Tsylova E. 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De evolvenda functiones (yd.yd.yd\u2026ydX\/n) disquisitiones nonullae analyticae Ph.D. thesis 1823 G\u0151ttinger Digitalisierungszentrum Berlin Germany."},{"key":"e_1_2_6_13_2","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/BF02784783","article-title":"Combinatorial aspects of boson algebra","volume":"10","author":"Katriel J.","year":"1974","journal-title":"Lettere al Nuovo Cimento"},{"key":"e_1_2_6_14_2","article-title":"On generalizations of the Stirling number triangles","volume":"3","author":"Lang W.","year":"2000","journal-title":"Journal of Integer Sequences"},{"key":"e_1_2_6_15_2","article-title":"Combinatorial interpretation of generalized Stirling numbers","volume":"12","author":"Lang W.","year":"2009","journal-title":"Journal of Integer Sequences"},{"key":"e_1_2_6_16_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(03)00194-4"},{"key":"e_1_2_6_17_2","doi-asserted-by":"publisher","DOI":"10.2307\/2371100"},{"key":"e_1_2_6_18_2","unstructured":"BlasiakP. 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