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Dae san Kim

Publications and source records attributed to Dae san Kim.

At least 37 records · Page 2Linked 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.

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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.

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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.

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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.

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arXiv2022

Dimorphic Mersenne numbers and their applications

The Mersenne primes are primes which can be written as some prime power of 2 minus 1. These primes were studied from antiquity in that their close connection with perfect numbers and even to present day in that their easiness for primality test. In this paper, we introduce the dimorphic Mersenne numbers as a degenerate version of the Mersenne numbers and investigate some of their properties in connection with the degenerate Bernoulli polynomials and the incomplete Bell polynomials.

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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.

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arXiv2022

Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via Boson operators

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r-Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. 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. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r-Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r-Dowling polynomials as a natural extension of the degenerate r-Whitney numbers of the second kind and investigate their properties.

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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.

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arXiv2022

Some identities involving degenerate Stirling numbers arising from normal ordering

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal oderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and from the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.

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arXiv2022

Some results on degenerate Fubini and degenerate Bell polynomials

The aim of this paper is to further study some properties and identities on the degenerate Fubini and the degenerate Bell polynomials which are degenerate versions of the Fubini and the Bell polynomials, respectively. Especially, we find several expressions for the generating function of the sum of the values of the generalized falling factorials at positive consecutive integers.

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arXiv2022

Central factorial numbers associated with sequences of polynomials

Many important special numbers appear in the expansions of some polynomials in terms of central factorials and vice versa, for example central factorial numbers, degenerate central factorial numbers, and central Lah numbers which are recently introduced. Here we generalize this to any sequence of polynomials. Let P be the space of a sequence of polynomials such that the degree of the polynomials less than n . The aim of this paper is to study the central factorial numbers of the second associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with P enjoy orthogonality and inverse relations. We illustrate our results with many examples and obtain interesting orthogonality and inverse relations by applying such relations for the central factorial numbers associated with P to each of our examples.

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arXiv2022

Stirling numbers associated with sequences of polynomials

Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. Our results are illustrated with many examples which give rise to interesting inverse relations in each case.

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arXiv2022

Some identities involving degenerate r-Stirling numbers

Recently, authors studied the unsigned degenerate r-Stirling number of the first kind and the degenerate r-Stirling number of the second kind, respectively of which are the degenerate versions of the unsigned r-Stirling numbers of the first kind and those of the r-Stirling numbers of the second kind. The aim of this paper is to derive some identities involving such special numbers from the inverse relations for them.

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arXiv2022

Some identities and properties on degenerate Stirling numbers

The aim of this paper is by using generating functions to further study some identities and properties on the degenerate Stirling numbers of the second kind, the degenerate $r$-Stirling numbers of the second kind, the degenerate Stirling numbers of the first kind and the unsigned degenerate Stirling numbers of the first kind.

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arXiv2021

Representations by ordered Bell and degenerate ordered Bell polynomials

In this paper, we consider the problem of representing any polynomial in terms of the ordered Bell and degenerate ordered Bell polynomials, and more generally of the higher-order ordered Bell and higher-order degenerate ordered Bell polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

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arXiv2021

Representing polynomials by degenerate Bernoulli polynomials

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials and more generally of the higher-order degenerate Bernoulli polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

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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.

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