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Taekyun Kim

Publications and source records attributed to Taekyun Kim.

At least 91 records · Page 5Linked to original sources
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

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 formulas for fully degenerate Bernoulli numbers and polynomials

The aim of this paper is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp. We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r-Stirling numbers of the second kind and of the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.

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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

A finite sum involving generalized falling factorial polynomials and degenerate Eulerian polynomials

The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the degenerate Bernoulli polynomials and the degenerate Frobenius-Euler polynomials. Secondly, we consider the degenerate Eulerian polynomials and deduce the generating function and a recurrence relation for them.

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arXiv2021

A Series transformation formula and related degenerate polynomials

Recently, Bovadzhiev studied a power series whose coefficients are binomial expressions and extended some known formulas involving classical special functions and polynomials. The aim of this paper is to adopt his ideas to express several identities involving " degenerate formal power series" as those including degenerate stirling numbers of the second kind, degenerate Bell polynomials, degenerate Fubini polynomials and degenerate poly-Bernoulli polynomials.

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arXiv2021

Some identities on degenerate Bell polynomials and their related identities

Recently, several types of degenerate Bell polynomials have been introduced as degenerate versions of the ordinary Bell polynomials. The aim of this paper is to study some identities for the degenerate Bell polynomials and their related identities by making use of series transformation formula obtained by Boyadzhiev. In particular, this answers to the natural question about the relationship between two different types of degenerate Bell polynomials.

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arXiv2021

On Some Summation Formulas

In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals

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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

Representation by degenerate Frobenius-Euler polynomials

The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

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arXiv2021

Some identities of fully degenerate Dowling and fully degenerate Bell polynomials arising from lambda-umbral calculus

Recently, Kim-Kim introduced the lambda-umbral calculus, in which the lambda-Sheffer sequences occupy the central position. In this paper, we introduce the fully degenerate Bell and the fully degenerate Dowling polynomials, and investigate some properties and identities relating to those polynomials with the help oflambda-umbral calculus. Here we note that the fully degenerate Bell poynomials and the fully degenerate Dowling polynomials are respectively degenerate versions of the Bell polynomials and the Dowling polynomials, of which the latters are the natural extension of the Whitney numbers of the second kind.

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arXiv2021

Some new properties on degenerate Bell polynomials

The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate version of the Bell numbers and polynomials. we derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of the both kinds.

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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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