New Formulas and Numbers Arising from Analyzing Combinatorial Numbers and Polynomials

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dc.contributor.authorKucukoglu, Irem
dc.contributor.authorŞimşek, Yilmaz
dc.date.accessioned2026-01-24T12:20:58Z
dc.date.available2026-01-24T12:20:58Z
dc.date.issued2021
dc.departmentAlanya Alaaddin Keykubat Üniversitesi
dc.description.abstractIn this paper, we derive various identities involving the negative higher-order combinatorial numbers and polynomials and other kinds of special numbers and polynomials such as the Stirling numbers, the Lah numbers, the negative higher-order Changhee numbers and polynomials, and the positive higher-order Bernoulli numbers and polynomials. Furthermore, by using the integral formulas of not only the negative higher-order combinatorial numbers and polynomials but also their generating functions, we obtain some identities and combinatorial sums. We give some infinite series, involving the negative higher-order combinatorial numbers, with their values in terms of the falling factorials, the Catalan numbers, the Daehee numbers (linear combination of the Stirling numbers and the Bernoulli numbers) and the Changhee numbers (linear combination of the Stirling numbers and the Euler numbers). As application of these infinite series, we also set two new sequences of special numbers with their generating functions, and investigate their properties. We pose an open question related to one of these number sequences. By using an infinite series arising from the integral of the generating functions for the negative higher-order combinatorial numbers and polynomials, we also introduce a new family of polynomials associated with the Bernstein basis functions. In addition, we derive symmetry property, integral formulas and derivative formula for these newly introduced polynomials. Moreover, by implementing an explicit formula of these newly introduced polynomials in Mathematica with the aid of the Wolfram programming language, we present some plots of these newly introduced polynomial functions for some of their randomly selected special cases. We also give some further results including series representations, combinatorial sums, integral formulas and relations for some of combinatorial numbers and poynomials. Finally, we present some observations and comments on our results. © 2021, MTJPAM Turkey. All rights reserved.
dc.identifier.endpage259
dc.identifier.issue3
dc.identifier.scopus2-s2.0-85112767210
dc.identifier.scopusqualityQ1
dc.identifier.startpage238
dc.identifier.urihttps://hdl.handle.net/20.500.12868/4753
dc.identifier.volume3
dc.indekslendigikaynakScopus
dc.language.isoen
dc.publisherMTJPAM Turkey
dc.relation.ispartofMontes Taurus Journal of Pure and Applied Mathematics
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanı
dc.rightsinfo:eu-repo/semantics/closedAccess
dc.snmzKA_Scopus_20260121
dc.subjectBell numbers
dc.subjectBernoulli numbers and polynomials
dc.subjectBernstein basis functions
dc.subjectCatalan numbers
dc.subjectDaehee and Changhee numbers
dc.subjectGenerating functions
dc.subjectLah numbers
dc.subjectSpecial numbers and polynomials
dc.subjectStirling numbers
dc.titleNew Formulas and Numbers Arising from Analyzing Combinatorial Numbers and Polynomials
dc.typeArticle

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