Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling
Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some p...
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MDPI AG
2023-07-01
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| Series: | Symmetry |
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| author | Noor Alam Waseem Ahmad Khan Can Kızılateş Sofian Obeidat Cheon Seoung Ryoo Nabawia Shaban Diab |
| author_facet | Noor Alam Waseem Ahmad Khan Can Kızılateş Sofian Obeidat Cheon Seoung Ryoo Nabawia Shaban Diab |
| author_sort | Noor Alam |
| collection | DOAJ |
| description | Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. |
| first_indexed | 2024-03-11T00:36:07Z |
| format | Article |
| id | doaj.art-51ae9dbfd28147118bbdd07f5f795285 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T00:36:07Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-51ae9dbfd28147118bbdd07f5f7952852023-11-18T21:33:57ZengMDPI AGSymmetry2073-89942023-07-01157135810.3390/sym15071358Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer ModelingNoor Alam0Waseem Ahmad Khan1Can Kızılateş2Sofian Obeidat3Cheon Seoung Ryoo4Nabawia Shaban Diab5Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, 67100 Zonguldak, TurkeyDepartment of Basic Sciences, Deanship of Preparatory Year, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Mathematics, Hannam University, Daejeon 34430, Republic of KoreaDepartment of Basic Sciences, Deanship of Preparatory Year, University of Ha’il, Ha’il 2440, Saudi ArabiaMany properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.https://www.mdpi.com/2073-8994/15/7/1358Changhee–Genocchi polynomialsChanghee–Frobenius-Euler polynomialsChanghee–Frobenius–Genocchi polynomials and numbers |
| spellingShingle | Noor Alam Waseem Ahmad Khan Can Kızılateş Sofian Obeidat Cheon Seoung Ryoo Nabawia Shaban Diab Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling Symmetry Changhee–Genocchi polynomials Changhee–Frobenius-Euler polynomials Changhee–Frobenius–Genocchi polynomials and numbers |
| title | Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling |
| title_full | Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling |
| title_fullStr | Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling |
| title_full_unstemmed | Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling |
| title_short | Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling |
| title_sort | some explicit properties of frobenius euler genocchi polynomials with applications in computer modeling |
| topic | Changhee–Genocchi polynomials Changhee–Frobenius-Euler polynomials Changhee–Frobenius–Genocchi polynomials and numbers |
| url | https://www.mdpi.com/2073-8994/15/7/1358 |
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