Searcharxiv⌕ Search

arXiv subjects

Hye Kyung Kim

Publications and source records attributed to Hye Kyung Kim.

17 recordsLinked to original sources
arXiv2023

New approach to $λ$-stirling numbers

The aim of this paper is to study the $λ$-Stirling numbers of both kinds which are $λ$-analogues of Stirling numbers of both kinds. Those numbers have nice combinatorial interpretations when $λ$ are positive integers. If $λ$ =1, then the $λ$-Stirling numbers of both kinds reduce to the Stirling numbers of both kinds. We derive new types of generating functions of the $λ$-Stirling numbers of both kinds which are related to the reciprocals of the generalized rising factorials. Furthermore, some related identities are also derived from those generating functions. In addition, all the corresponding results to the $λ$-Stirling numbers of both kinds are obtained also for the $λ$-analogues of r-Stirling numbers of both kinds which are generalizations of those numbers.

math.NT↗
arXiv2023

Generalized degenerate stirling numbers arising from degenerate boson normal ordering

It is remarkable that, in recent years, intensive studies have been done for degenerate versions of many special polynomials and numbers and have yielded many interesting results. The aim of this paper is to study the generalized degenerate (r, s)-Stirling numbers of the second and their natural extensions to polynomials, namely the generalized degenerate (r, s)-Bell polynomials, arising from certain degenerate boson normal ordering. We derive some properties, explicit expressions and generating functions for those numbers and polynomials. The generalized degenerate (r, s)-Stirling numbers of the second and the degenerate boson normal ordering are respectively degenerate versions of the generalized (r, s)-Stirling numbers of the second and the boson normal ordering studied earlier by Blasiak-Person-Solomon.

math.NT↗
arXiv2023

Study on discrete degenerate Bell distributions with two parameters

Recently, Freud-Rodriguez proposed a new counting process which is called the Bell-Touchard process and based on the Bell-Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are 'degenerate versions' of the Bell-Touchard probability distributions and the Bell-Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process.

math.NT↗
arXiv2023

Some identities on degenerate hyperbolic functions arising from $p$-adic integrals on $\mathbb{Z}_p$

The aim of this paper is to introduce several degenerate hyperbolic functions as degenerate versions of the hyperbolic functions, to evaluate Volkenborn and the fermionic $p$-adic integrals of the degenerate hyperbolic cosine and the degenerate hyperbolic sine functions and to derive from them some identities involving the degenerate Bernoulli numbers, the degenerate Euler numbers and the Cauchy numbers of the first kind.

math.NT↗
arXiv2023

Multi-Stirling numbers of the second kind

The multi-Stirling numbers of the second kind, the unsigned multi-Stirling numbers of the first kind, the multi-Lah numbers and the multi-Bernoulli numbers are all defined with the help of the multiple logarithm, and generalize respectively the Stirling numbers of the second kind, the unsigned Stirling numbers of the first kind, the unsigned Lah numbers and the higher-order Bernoulli numbers . The aim of this paper is to introduce the multi-Stirling numbers of the second kind and to find several identities involving those four numbers defined by means of the multiple logarithm and some other special numbers.

math.NT↗
arXiv2023

A note on infinite series whose terms involve truncated degenerate exponentials

The degenerate exponentials play an important role in recent study on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with several special numbers and to find either their values or some other expressions of them as finite sums.

math.NT↗
arXiv2023

Identities involving degenerate harmonic and degenerate hyperharmonic numbers

Harmonic numbers have been studied since antiquity, while hyperharmonic numbers were intoduced by Conway and Guy in 1996. The degenerate harmonic numbers and degenerate hyperharmonic numbers are their respective degenerate versions. The aim of this paper is to further investigate some properties, recurrence relations and identities involving the degenerate harmonic and degenerate hyperharmonic numbers in connection with degenerate Stirling numbers of the first kind, degenerate Daehee numbers and degenerate derangements.

math.NT↗
arXiv2022

Some identities on generalized harmonic numbers and generalized harmonic functions

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory and analysis of algorithms. The aim of this paper is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F(x)= B( x+1, n+1), ( n=0,1,2,..) using elementary methods.

math.NT↗
arXiv2022

On generalized degenerate Euler-Genocchi polynomials

We introduce the generalized degenerate Euler-Genocchi polynomials as a degenerate version of the Euler-Genocchi polynomials. In addition, we introduce their higher-order version, namely the generalized degenerate Euler-Genocchi polynomials of order α, as a degenerate version of the generalized Euler-Genocchi polynomials of order α. The aim of this paper is to study certain properties and identities involving those polynomials, the generalized falling factorials, the degenerate Euler polynomials of order α, the degenerate Stirling numbers of the second kind, and the alternating degenerate power sum of integers.

math.NT↗
arXiv2022

Degenerate r-Bell polynomials arising from degenerate normal odering

Recently, Kim-Kim introduced the degenerate r-Bell polynomials and investigated some results which are derived from umbral calculus. The aim of this paper is to study some properties of the degenerate r-Bell polynomials and numbers via boson operators. In particular, we obtain two expressions for the generating function of the degenerate r-Bell polynomials in |z| , and a recurrence relation and Dobinski-like formula for the degenerate r-Bell numbers. These are derived from the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators where the degenerate r-Stirling numbers of the second kind appear as the coefficients.

math.NT↗
arXiv2022

Study on r-truncated degenerate stirling numbers of the second kind

The degenerate Stirling numbers of the second kind and of the first kind, which are respectively degenerate versions of the Stirling numbers of the second kind and of the first kind, appear frequently when we study various degenerate versions of some special numbers and polynomials. The aim of this paper is to consider the r-truncated degenerate Stirling numbers of the second kind, which reduce to the degenerate Stirling numbers of the second for r = 1, and to investigate their explicit expressions, some properties and related identities, in connection with several other degenerate special numbers and polynomials.

math.NT↗
arXiv2022

Some identities on $λ$-analogues of $r$-stirling numbers of the second kind

Recently, the $λ$-analogues of $r$-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the $λ$-analogues of $r$-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the $λ$-analogues of Whitney-type $r$-Stirling numbers of the second and derive similar results to the case of the $λ$-analogues of r-Stirling numbers of the second kind. In addition, we consider the $λ$-analogues of Dowling polynomials and deduce a Dobinski-like formula.

math.NT↗
arXiv2022

Normal ordering of degenerate integral powers of number operator and its applications

The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a `degenerate version' of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials.

math.NT↗
arXiv2021

A note on degenerate generalized Laguerre polynomials and Lah numbers

The 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.

math.CA↗
arXiv2020

A Computer-Assisted Study of Red Coral Population Dynamics

We consider a 13-dimensional age-structured discrete red coral population model varying with respect to a fitness parameter. Our numerical results give a bifurcation diagram of both equilibria and stable invariant curves of orbits. We observe that not only for low levels of fitness, but also for high levels of fitness, populations are extremely vulnerable, in that they spend long time periods near extinction. We then use computer-assisted proofs techniques to rigorously validate the set of regular and bifurcation fixed points that have been found numerically.

math.DS↗
arXiv2007

Generalized characteristic polynomials of graph bundles

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.

math.CO↗