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Inspired by their work, we consider a new class of the Frobenius\u2013Genocchi polynomials, which is called the type 2 poly-Frobenius\u2013Genocchi polynomials, by means of the polyexponential function. We also derive some new relations and properties including the Stirling numbers of the first and second kinds. In a special case, we give a relation between the type 2 poly-Frobenius\u2013Genocchi polynomials and Bernoulli polynomials of order k<\/jats:italic>. Moreover, motivated by the definition of the unipoly-Bernoulli polynomials given in (Kim and Kim in Russ. J. Math. Phys. 26(1):40\u201349, 2019), we introduce the unipoly-Frobenius\u2013Genocchi polynomials via a unipoly function and give multifarious properties including derivative and integral properties. 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Furthermore, we provide a correlation between the unipoly-Frobenius\u2013Genocchi polynomials and the classical Frobenius\u2013Genocchi polynomials.","DOI":"10.1186\/s13662-020-02889-2","type":"journal-article","created":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T13:04:08Z","timestamp":1597755848000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Construction of the type 2 poly-Frobenius\u2013Genocchi polynomials with their certain applications"],"prefix":"10.1186","volume":"2020","author":[{"given":"Ugur","family":"Duran","sequence":"first","affiliation":[]},{"given":"Mehmet","family":"Acikgoz","sequence":"additional","affiliation":[]},{"ORCID":"http:\/\/orcid.org\/0000-0002-3950-6864","authenticated-orcid":false,"given":"Serkan","family":"Araci","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,18]]},"reference":[{"key":"2889_CR1","doi-asserted-by":"crossref","DOI":"10.1155\/2009\/382574","volume":"2009","author":"M. 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Abstract
Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.
MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
