Searcharxiv⌕ Search

arXiv subjects

Dae san Kim

Publications and source records attributed to Dae san Kim.

At least 19 recordsLinked to original sources
arXiv2026

A study of degenerate Bernoulli and Euler numbers via operators

The paper introduces novel classes of operators and investigates their applications to the study of special numbers. We utilize these operators to derive explicit expressions for degenerate Euler numbers and type 2 degenerate Euler numbers. Furthermore, we establish a relationship between degenerate Bernoulli numbers and degenerate Euler numbers. We note here that these operators emerge naturally when we study such explicit expressions and such a relationship

math.NT↗
arXiv2026

Study on Morgan-Voyce type polynomials with Euler-Seidel algorithm

This paper bridges the domains of degenerate special polynomials, the Euler-Seidel matrix method, and Morgan-Voyce polynomials. We introduce two new families of Morgan-Voyce type polynomials and establish their structural properties, including explicit formulas and recurrence relations. Additionally, we define three polynomial variants and prove that three distinct binomial-type sums for Bell polynomials can be expressed as finite sums involving these new families, and derive their exponential generating functions. We then construct Euler-Seidel matrices using the initial sequences associated with Morgan-type polynomials. Our results yield novel algebraic identities and expand the application of matrix methods in combinatorial analysis.

math.NT↗
arXiv2026

Generalized Bell polynomial operators arising from generalized normal ordering

This paper explores the deformed combinatorial structures arising from the generalized Heisenberg algebra GHA which is characterized by an analytic function of the Hamiltonian f(H) and governs systems with non-linear spectra. Moving beyond the classical Heisenberg-Weyl framework, we investigate the normal ordering of the generalized number operator $N_f^n = (\fa)^n$, which naturally introduces the generalized Stirling operators of the second kind. Using the vacuum eigenvalue the Hamiltonian, we define quantum operator factorials and a generalized quantum exponential function. We explicitly construct the generalized coherent states and derive several operator identities. Notably, we prove that the powers of quantum operators expand into generalized falling factorials, and that the coherent state expectation values are explicitly given by the generalized Bell polynomial operators.

math.NT↗
arXiv2026

Logarithms and Stirling numbers associated with delta series

This paper investigates the Stirling numbers of the first and second kind associated with a delta series f (t). These numbers provide a robust framework that satisfies the orthogonality and inverse relations, often lacking in recent probabilistic Stirling and B-Stirling numbers. Key contributions include the definition and analysis of the logarithm associated with a delta series f (t). We further establish a Schlomilch-type formula, which provides an explicit connection between the two kinds of Stirling numbers. Using this formula, we derive another expression for the associated logarithm in terms of the Stirling numbers of the second kind associated with a delta series f (t). Finally, we provide fifteen concrete examples to illustrate the versatility of this framework, demonstrating how it unifies and extends several known results in combinatorial analysis.

math.NT↗
arXiv2026

Degenerate Algorithms for degenerate Bernoulli and Euler numbers

This paper introduces and investigates degenerate versions of the A-algorithm and B-algorithm by incorporating a parameter lambda into their respective recurrence relations. We derive explicit formulas for the final sequences of these algorithms in terms of the initial sequences and the degenerate Stirling numbers of the second kind. Furthermore, we establish functional relationships between the ordinary generating functions of the initial sequences and the exponential generating functions of the final sequences. Specifically, we demonstrate that these degenerate algorithms yield degenerate Bernoulli and Euler numbers under specific initial conditions.

math.NT↗
arXiv2025

Degenerate Euler-Seidel Matrix Method and Their Applications

This paper introduces a degenerate version of the Euler-Seidel matrix method by incorporating a parameter lambda into the classical recurrence relation. The standard Euler-Seidel method relates the generating functions of an initial sequence and its final sequence via Seidel's formula, Our generalized method establishes transformation formulas using lambda-generalized binomial identities and yields a degenerate Seidel's formula for the exponential generating functions. The results are applied to study and derive new combinatorial identities for sequences like the degenerate Bell and Fubini numbers and polynomials.

math.NT↗
arXiv2025

Representations by probabilistic Bernoulli and degenerate Bernoulli polynomials

We investigate the representation of arbitrary polynomials using probabilistic Bernoulli and degenerate Bernoulli polynomials associated with a random variable $Y$, whose moment generating function exists in a neighborhood of the origin. In addition, this paper explores the problem of representing arbitrary polynomials in terms of their higher-order counterparts. We develop explicit formulas for those representations with the help of umbral calculus and illustrate our results for several discrete and continuous random variables Y.

math.NT↗
arXiv2025

Recurrence relations for harmonic and derangement numbers

We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the underlying structure of these numbers but also highlight the effectiveness of elementary techniques in discovering new mathematical properties. The findings have potential applications in various fields where these numbers appear, including combinatorics, probability, and computer science.

math.NT↗
arXiv2025

A note on new type degenerate Srirling numbers of the first kind

We introduce a new sequence of unsigned degenerate Stirling numbers of the first kind. Following the work of Adell-Lekuona, who represented unsigned Stirling numbers of the first kind as multiples of the expectations of specific random variables, we express our new numbers as finite sums of multiples of the expectations of certain random variables. We also provide a representation of these new numbers as finite sums involving the classical unsigned Stirling numbers of the first kind. As an inversion formula, we define a corresponding sequence of new type degenerate Stirling numbers of the second kind. We derive expressions for these numbers as finite sums that involve the Stirling numbers of the second kind.

math.NT↗
arXiv2025

Several expressions for degenerate harmonic numbers and some related numbers

Many authors have recently studied the degenerate harmonic numbers. This paper makes two main contributions. First, we derive several explicit expressions for these numbers, which are a degenerate version of the ordinary harmonic numbers. We also examine the degenerate harmonic numbers of order m and find an expression for them. Second, we investigate some related numbers that are closely connected to the degenerate harmonic numbers of order m, which reduce to the degenerate harmonic numbers when m=1

math.NT↗
arXiv2025

Degenerate Sheffer-type polynomials and degenerate Sheffer polynomials associated with a random variable

This paper has two primary contributions. First, we explore degenerate Sheffer-type polynomials, a hybrid of higher-order degenerate Bernoulli and Euler polynomials, and derive their properties. Second, assuming that the moment generating function of Y exists in a neighborhood of the origin, we introduce the degenerate Sheffer polynomials associated with Y. We then investigate their properties in general and for the specific cases of uniform and Bernoulli random variables. We also present new results for the higher-order degenerate Bernoulli and Euler polynomials.

math.NT↗
arXiv2025

Combinatorial identities related to degenerate Stirling numbers of the second kind

The study of degenerate versions of certain special polynomials and numbers, which was initiated by Carlitz's work on degenerate Euler and degenerate Bernoulli polynomials, has recently seen renewed interest among mathematicians. The aim of this paper is to study some properties, certain identities, recurrence relations and explicit expressions for degenerate Stirling numbers of the second kind, which are a degenerate version of the Stirling numbers of the second kind. These numbers appear very frequently when we study various degenerate versions of many special polynomials and numbers. Especially, we consider some closely related polynomials and power series in connection with a degenerate version of Euler's formula for the Stirling numbers of the second kind.

math.NT↗
arXiv2024

Degenerate Eulerian polynomials and numbers

The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties. Specifically, we derive an identity, recursive relations, generating function and degenerate version of Worpitzky's identity for the degenerate Eulerian polynomials and numbers. In addition, we obtain several results involving the degenerate Stirling numbers of the second kind and the degenerate Bernoulli numbers as well as the degenerate Eulerian numbers.

math.NT↗
arXiv2024

Probabilistic multi-Stirling numbers of the second kind and probabilistic multi-Lah numbers

Assume that the moment generating function of the random vari able Y exists in a neighborhood of the origin. We introduce the probabilistic multi-Stirling numbers of the second kind associated with Y and the proba bilistic multi-Lah numbers associated with Y, both of indices (k1,k2,...,kr), by means of the multiple logarithm. Those numbers are respectively probabilistic extensions of the multi-Stirling numbers of the second kind and the multi-Lah numbers which, for (k1,k2,...,kr) = (1,1,...,1), boil down respectively to the Stirling numbers of the second and the unsigned Lah numbers. The aim of this paper is to study some properties, related identities, recurrence relations and explicit expressions of those probabilistic extension numbers in connection with several other special numbers

math.NT↗
arXiv2024

Probabilistic degenerate Stirling numbers of the first kind and their applications

Let Y be a random variable whose degenerate moment generating functions exist in some neighborhoods of the origin. The aim of this paper is to study the probabilistic degenerate Stirling numbers of the first kind associated with Y which are constructed from the degenerate cumulant generating function of Y. They are a degenerate version of the probabilistic Stirling numbers of the first kind associated with Y, which were recently introduced by Adell-Benyi. We investigate some properties, related identities, recurrence relations and explicit expressions for those numbers. In addition, we apply our results to the special cases of normal and gamma random variables.

math.CO↗
arXiv2024

Probabilistic bivariate Bell polynomials

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. We consider the probabilistic bivariate Bell polynomials associated with Y and the probabilistic bivariate r-Bell polynomials associated with Y. For those polynomials, we derive the recurrence relations corresponding to the ones found by Zheng and Li for the bivariate Bell polynomials and the bivariate r-Bell polynomials.

math.NT↗