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Search Results (3)

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Keywords = degenerate λ-Stirling polynomials of the second kind

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14 pages, 282 KiB  
Article
A Note on Degenerate Catalan-Daehee Numbers and Polynomials
by Waseem Ahmad Khan, Maryam Salem Alatawi and Ugur Duran
Symmetry 2022, 14(10), 2169; https://doi.org/10.3390/sym14102169 - 16 Oct 2022
Cited by 4 | Viewed by 1506
Abstract
In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of λ-Daehee numbers, Stirling numbers [...] Read more.
In this paper, we consider the degenerate forms of the Catalan–Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan–Daehee numbers in terms of λ-Daehee numbers, Stirling numbers of the first kind and Bernoulli polynomials, and we also obtain a relation covering the Bernoulli numbers, the degenerate Catalan–Daehee numbers and Stirling numbers of the second kind. In addition, we prove an implicit summation formula and a symmetric identity, and we derive an explicit expression for the degenerate Catalan–Daehee polynomials including the Stirling numbers of the first kind and Bernoulli polynomials. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)
14 pages, 296 KiB  
Article
Degenerate Derangement Polynomials and Numbers
by Minyoung Ma and Dongkyu Lim
Fractal Fract. 2021, 5(3), 59; https://doi.org/10.3390/fractalfract5030059 - 22 Jun 2021
Cited by 3 | Viewed by 2299
Abstract
In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. [...] Read more.
In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
11 pages, 236 KiB  
Article
Degenerate Stirling Polynomials of the Second Kind and Some Applications
by Taekyun Kim, Dae San Kim, Han Young Kim and Jongkyum Kwon
Symmetry 2019, 11(8), 1046; https://doi.org/10.3390/sym11081046 - 14 Aug 2019
Cited by 34 | Viewed by 3899
Abstract
Recently, the degenerate λ -Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate λ -Stirling polynomials as well as the r-truncated degenerate λ -Stirling polynomials of the [...] Read more.
Recently, the degenerate λ -Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate λ -Stirling polynomials as well as the r-truncated degenerate λ -Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate λ -Stirling polynomials of the second and the r-truncated degenerate λ -Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. Full article
(This article belongs to the Special Issue The 32th Congress of The Jangjeon Mathematical Society (ICJMS2019) will be Held at Far Eastern Federal Universit, Vladivostok Russia)
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