scholarly journals Escaping from a degenerate version of the four hill potential

2019 ◽  
Vol 126 ◽  
pp. 12-22
Author(s):  
Euaggelos E. Zotos ◽  
Wei Chen ◽  
Christof Jung
Keyword(s):  
Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1086 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Hyunseok Lee ◽  
Jongkyum Kwon

Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind.


Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 507-514
Author(s):  
Dmitry Dolgy ◽  
Dae Kim ◽  
Taekyun Kim ◽  
Jongkyum Kwon

Recently, the partially degenerate Bell numbers and polynomials were introduced as a degenerate version of Bell numbers and polynomials. In this paper, as a further degeneration of them, we study fully degenerate Bell numbers and polynomials. Among other things, we derive various expressions for the fully degenerate Bell numbers and polynomials.


2021 ◽  
Vol 7 (3) ◽  
pp. 3845-3865
Author(s):  
Hye Kyung Kim ◽  
◽  
Dmitry V. Dolgy ◽  

<abstract><p>Many mathematicians have studied degenerate versions of some special polynomials and numbers that can take into account the surrounding environment or a person's psychological burden in recent years, and they've discovered some interesting results. Furthermore, one of the most important approaches for finding the combinatorial identities for the degenerate version of special numbers and polynomials is the umbral calculus. The Catalan numbers and the Daehee numbers play important role in connecting relationship between special numbers.</p> <p>In this paper, we first define the degenerate Catalan-Daehee numbers and polynomials and aim to study the relation between well-known special polynomials and degenerate Catalan-Daehee polynomials of order $ r $ as one of the generalizations of the degenerate Catalan-Daehee polynomials by using the degenerate Sheffer sequences. Some of them include the degenerate and other special polynomials and numbers such as the degenerate falling factorials, the degenerate Bernoulli polynomials and numbers of order $ r $, the degenerate Euler polynomials and numbers of order $ r $, the degenerate Daehee polynomials of order $ r $, the degenerate Bell polynomials, and so on.</p></abstract>


Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 40 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hye Kyung Kim

AbstractUmbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim–Kim (J. Math. Anal. Appl. 493(1):124521, 2021) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined the λ-linear functionals and λ-differential operators, respectively, instead of the linear functionals and the differential operators of umbral calculus established by Rota. In this paper, the author gives various interesting identities related to the degenerate Lah–Bell polynomials and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derives the inversion formulas of these identities.


1999 ◽  
Vol 09 (03n04) ◽  
pp. 405-430 ◽  
Author(s):  
DANIEL KROB ◽  
JEAN-YVES THIBON

We interpret quasi-symmetric functions and noncommutative symmetric functions as characters of a degenerate quantum group obtained by putting q=0 in a variant of Uq(glN).


Author(s):  
Jacques Durand

Corpus Phonology is an approach to phonology that places corpora at the center of phonological research. Some practitioners of corpus phonology see corpora as the only object of investigation; others use corpora alongside other available techniques (for instance, intuitions, psycholinguistic and neurolinguistic experimentation, laboratory phonology, the study of the acquisition of phonology or of language pathology, etc.). Whatever version of corpus phonology one advocates, corpora have become part and parcel of the modern research environment, and their construction and exploitation has been modified by the multidisciplinary advances made within various fields. Indeed, for the study of spoken usage, the term ‘corpus’ should nowadays only be applied to bodies of data meeting certain technical requirements, even if corpora of spoken usage are by no means new and coincide with the birth of recording techniques. It is therefore essential to understand what criteria must be met by a modern corpus (quality of recordings, diversity of speech situations, ethical guidelines, time-alignment with transcriptions and annotations, etc.) and what tools are available to researchers. Once these requirements are met, the way is open to varying and possibly conflicting uses of spoken corpora by phonological practitioners. A traditional stance in theoretical phonology sees the data as a degenerate version of a more abstract underlying system, but more and more researchers within various frameworks (e.g., usage-based approaches, exemplar models, stochastic Optimality Theory, sociophonetics) are constructing models that tightly bind phonological competence to language use, rely heavily on quantitative information, and attempt to account for intra-speaker and inter-speaker variation. This renders corpora essential to phonological research and not a mere adjunct to the phonological description of the languages of the world.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Hanyoung Kim

AbstractA new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Hye Kyung Kim ◽  
Seong Ho Park

AbstractThe aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues type formula, and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.


2003 ◽  
Vol 34 (2) ◽  
pp. 160-198 ◽  
Author(s):  
Katell Berthelot

AbstractIn this article I have tried to show that both Strabo's text (16.2.34-46) and Diodorus' text (34-35.1.1-5) about the Jews can be attributed to Posidonius of Apamea and that they do not contradict each other; nor do they contradict Josephus' testimony in the Against Apion (2.79), where it is very difficult anyway to determine what exactly goes back to Posidonius. His vision of the Jews can be summarized as follows: although he considered Moses as a wise and pious man who founded an admirable political and religious community, for political and philosophical reasons Posidonius greatly despised and disliked the Judaism of his time, which he regarded as a degenerate version of the Mosaic project.


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