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Keywords = Changhee–Genocchi polynomials

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20 pages, 1302 KiB  
Article
Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling
by Noor Alam, Waseem Ahmad Khan, Can Kızılateş, Sofian Obeidat, Cheon Seoung Ryoo and Nabawia Shaban Diab
Symmetry 2023, 15(7), 1358; https://doi.org/10.3390/sym15071358 - 4 Jul 2023
Cited by 10 | Viewed by 1479
Abstract
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 [...] Read more.
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 α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. Full article
(This article belongs to the Special Issue Special Functions and Orthogonal Polynomials: Symmetry and Applications)
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12 pages, 258 KiB  
Article
A Note on Modified Degenerate Changhee–Genocchi Polynomials of the Second Kind
by Waseem Ahmad Khan and Maryam Salem Alatawi
Symmetry 2023, 15(1), 136; https://doi.org/10.3390/sym15010136 - 3 Jan 2023
Cited by 9 | Viewed by 1399
Abstract
In this study, we introduce modified degenerate Changhee–Genocchi polynomials of the second kind, and analyze some properties by providing several relations and applications. We first attain diverse relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier [...] Read more.
In this study, we introduce modified degenerate Changhee–Genocchi polynomials of the second kind, and analyze some properties by providing several relations and applications. We first attain diverse relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. By using their generating function, we derive some new relations, including the Stirling numbers of the first and second kinds. Moreover, we introduce modified higher-order degenerate Changhee–Genocchi polynomials of the second kind. We also derive some new identities and properties of this type of polynomials. Full article
(This article belongs to the Special Issue Application of Symmetry/Asymmetry in Fractional Differential Equations)
14 pages, 241 KiB  
Article
A Note on Higher Order Degenerate Changhee–Genocchi Numbers and Polynomials of the Second Kind
by Liwei Liu and Wuyungaowa
Symmetry 2023, 15(1), 56; https://doi.org/10.3390/sym15010056 - 26 Dec 2022
Viewed by 1187
Abstract
In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. [...] Read more.
In this paper, we consider the higher order degenerate Changhee–Genocchi polynomials of the second kind by using generating functions and the Riordan matrix methods. At the same time, we give some properties of the higher order degenerate Changhee–Genocchi polynomials of the second kind. In addition, we establish some new equalities involving the higher order degenerate Changhee–Genocchi polynomials of the second kind, the generalized Bell polynomials, higher order Changhee polynomials, the higher order degenerate Daehee polynomials of the second kind, Lah numbers and Stirling numbers, etc. Full article
11 pages, 254 KiB  
Article
New Type of Degenerate Changhee–Genocchi Polynomials
by Maryam Salem Alatawi and Waseem Ahmad Khan
Axioms 2022, 11(8), 355; https://doi.org/10.3390/axioms11080355 - 23 Jul 2022
Cited by 15 | Viewed by 1658
Abstract
A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim. We investigate some properties of these [...] Read more.
A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim. We investigate some properties of these numbers and polynomials. We also introduce a higher-order new type of degenerate Changhee–Genocchi numbers and polynomials which can be represented in terms of the degenerate logarithm function. Finally, we derive their summation formulae. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)
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