arXiv subjects

Dae San Kim

Publications and source records attributed to Dae San Kim.

At least 19 recordsLinked to original sources

arXiv2026

On Generalized Chebyshev polynomials

In this paper, we introduce and study a novel family of generalized Chebyshev polynomials defined via a rational generating function. We demonstrate that the classical Chebyshev polynomials of the first, second, third, and fourth kinds naturally emerge as special cases of this framework. Furthermore, we derive comprehensive recurrence relations, tridiagonal determinant representations, and explicit closed-form expressions for these polynomials. We also establish some connections between the generalized Chebyshev polynomials and other classical sequences, including Morgan-Voyce polynomials, Fibonacci polynomials, and Fubini polynomials via Stirling numbers of the second kind. Finally, we examine the associated Euler-Seidel matrix and provide a continued fraction expansion for the quotient of consecutive polynomial terms.

math.NT↗

arXiv2026

Degenerate generalized Stirling operators of the first kind arising from generalized Heisenberg algebra

This paper investigates the degenerate generalized Stirling operators of the first kind bridging a gap in the operational calculus of the generalized Heisenberg algebra GHA unified with degenerate calculus. As they are the inverse of the degenerate generalized Stirling operators of the second kind, these operators express the monomial operator products in terms of the degenerate factorial operators. We derive key structural and combinatorial properties for these operators, including an explicit product factorization, a fundamental recurrence relation, and an operational shifting identity. Furthermore, we establish the orthogonality relations between the degenerate generalized Stirling operators of the first and second kinds, providing a complete combinatorial framework for functional quantum algebras.

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arXiv2026

Probabilistic degenerate logarithm and heterogeneous stirling numbers

Let Y be a random variable whose moment-generating function exists in some neighborhood of the origin. While probabilistic Stirling numbers of the first and second kind have been introduced, early definitions often failed to satisfy fundamental orthogonality and inverse relations or lacked consistency with classical forms in the case when Y = 1. This paper addresses these limitations by utilizing redefined probabilistic Stirling numbers of the first kind and the second kind alongside their degenerate counterparts. Our primary objective is twofold: first,to introduce the probabilistic (degenerate) logarithm associated with Y, providing explicit expressions for various random variables and defining new probabilistic degenerate Daehee and Cauchy numbers; and second, to investigate probabilistic heterogeneous Stirling numbers and establish a probabilistic degenerate version of the Schlomilch formula, demonstrating that these new frameworks maintain the essential algebraic properties of their classical counterparts.

math.NT↗

arXiv2026

Probabilistic heterogeneous Stirling numbers and Bell polynomials

Let Y be a random variable satisfying specific moment conditions. This paper introduces and investigates probabilistic heterogeneous Stirling numbers of the second kind and probabilistic heterogeneous Bell polynomials. These structures unify several classical and probabilistic families, including those of Stirling, Lah, Bell and Lah-Bell. By integrating the heterogeneous framework of Kim and Kim with probabilistic extensions, we derive explicit formulas, Dobiński-like identities, and recurrence relations. We further establish connections to partial Bell polynomials and provide applications for Poisson and Bernoulli distributions.

math.NT↗

arXiv2025

Degenerate Euler- Seidel Method for degenerate Bernoulli, Euler, and Genocchi polynomials

This paper introduces a degenerate version of the Euler-Seidel method by incorporating a parameter lambda into the classical recurrence relation. We define a degenerate Euler-Seidel matrix associated with an initial sequence and establish corresponding lambda-generalized binomial identities and generating function relations. By applying this method to the degenerate Bernoulli, Euler, and Genocchi polynomials, we derive several new combinatorial identities. This work extends the classical Euler-Seidel method to the domain of degenerate special polynomials and numbers, providing a new framework for studying their properties.

math.NT↗

arXiv2025

Probabilistic Stirling numbers associated with sequences

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. Recently, probabilistic Stirling numbers of the first kind and of the second kind associated with Y have been introduced. However, probabilistic stirling number of the first kind based on the cumulant generating function of Y, and probabilistic stirling number of the second kind do not satisfy orthogonality and inverse relations. This paper aims to redefine the probabilistic stirling numbers of the first kind associated with Y such that probabilistic stiirling number of the first kind and probabilistic stirling number of the second kind do satisfy these crucial relations. Furthermore, we investigate their degenerate counterparts, the probabilistic degenerate Stirling numbers of both kinds. We explicitly compute probabilistic stiirling number of the first kind and probabilistic stirling number of the second kind .

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arXiv2025

A Generalized Recurrence for fully degenerate Bell polynomials

This paper addresses the unnatural appearance of the two-variable degenerate Fubini polynomials in a recently derived Spivey-type recurrence relation for the fully degenerate Bell polynomials. To solve this, we introduce a new family of polynomial which we also call the fully degenerate Bell polynomials, along with their two-variable counterparts. Our main contribution is the derivation of natural Spivey-type recurrence relations using operator methods. We extend these results to the r-counterparts, the fully degenerate r-Bell polynomials providing Dobinski-like, finite sum, operator expressions, and Spivey-type recurrence relations for all the new polynomials.

math.CO↗

arXiv2025

Spivey-type recurrence relation for fully degenerate Bell polynomials

Spivey's combinatorial method revealed an important identity for Bell numbers, involving Stirling numbers of the second kind. This paper extends his work by deriving Spivey-type recurrence relations for fully degenerate Bell polynomials and degenerate Fubini polynomials. Our derivation uses degenerate Stirling numbers of the second kind and two-variable degenerate Fubini polynomials of order a.

math.NT↗

arXiv2025

Representations by probabilistic Frobenius-Euler and degenerate Frobenius-Euler polynomials

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to represent arbitrary polynomials in terms of probabilistic Frobenius-Euler polynomials associated with Y and probabilistic degenerate Frobenius-Euler polynomials associated with Y , and more generally of their higher-order counterparts. We derive explicit formulas with the help of umbral calculus and illustrate our results in the case of several random variables Y

math.NT↗

arXiv2025

Probabilistic generalization of Spivey-type relation for degenerate Bell polynomials

Following Spivey's pivotal discovery of a recurrence relation for Bell numbers, significant research has emerged concerning various generalizations of Bell numbers and polynomials. For example, Kim and Kim established a Spivey-type recurrence relation specifically for degenerate Bell and Dowling polynomials. In this paper, we extend this work by deriving a probabilistic generalization of Spivey-type recurrence relations for both degenerate Bell and degenerate r-Bell polynomials.

math.NT↗

arXiv2025

Heterogeneous Stirling numbers and heterogeneous Bell polynomials

This paper introduces a novel generalization of Stirling and Lah numbers, termed ``heterogeneous Stirling numbers," which smoothly interpolate between these classical combinatorial sequences. Specifically, we define heterogeneous Stirling numbers of the second and first kinds, demonstrating their convergence to standard Stirling numbers for lambda=0 and to (signed) Lah numbers for lambda =1. We derive fundamental properties, including generating functions, explicit formulas, and recurrence relations. Furthermore, we extend these concepts to heterogeneous Bell polynomials, obtaining analogous results such as generating function, combinatorial identity and Dobinski-like formula. Finally, we introduce and analyse heterogeneous r-Stirling numbers of the second kind and their associated r-Bell polynomials.

math.GM↗

arXiv2025

Spivey's type recurrence relation for Lah-Bell polynomials

The aim of this paper is to derive Spivey's type recurrence relations for the Lah-Bell polynomials and the r-Lah-Bell polynomials by utilizing operators X and D satisfying the commutation relation DX-XD=1. Here X is the `multiplication by x' operator and D is the differentiation operator D=d/dx. In addition, we obtain Spivey's type recurrence relation for the lambda analogue of r-Lah-Bell polynomials by some other method without using the operators X and D.

math.CA↗

arXiv2025

Probabilistic degenerate poly-Bell polynomials associated with random variables

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study the probabilistic degenerate poly-Bell polynomials associated with the random variable Y, arising from the degenerate polyexponential functions, which are probabilistic extensions of degenerate versions of the poly-Bell polynomials. We derive several explicit expressions and some related identities for them. In addition, we consider the special cases that Y is the Bernoulli random variable with probability of success p or the gamma random variable with parameters 1,1.

math.NT↗

arXiv2025

Recurrence relations for degenerate Bell and Dowling polynomials via Boson operators

Spivey found a recurrence relation for the Bell numbers by using combinatorial method. The aim of this paper is to derive Spivey's type recurrence relations for the degenerate Bell polynomials and the degenerate Dowling polynomials by using the boson annihilation and creation operators satisfying the commutation relation aa+-a+a=1. In addition, we derive a Spivey's type recurrence relation for the r-Dowling polynomials.

math.NT↗

arXiv2025

Spivey's type recurrence relation for degenerate Bell polynomials

The aim of this paper is to derive a recurrence relation for the degenerate Bell polynomials by using the operators X and D satisfying the commutation relation DX-XD=1. Here X is the `multiplication by x' operator and D=d/dx. This recurrence relation is a generalization of Spivey's recurrence relation for the Bell numbers. We also obtain a recurrence relation for the degenerate r--Bell polynomials by using the same operators.

math.NT↗

arXiv2024

Probabilistic degenerate derangement polynomials

In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangement of an n-element set is called the nth derangement number. Recently, the degenerate derangement numbers and polynomials have been studied as degenerate versions. Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. In this paper, we study probabilistic extension of the degenerate derangement numbers and polynomials, namely the probabilistic degenerate derangement numbers and polynomials associated with Y. In addition, we consider the probabilistic degenerate r-derangement numbers associated with Y and the probabilistic degenerate derangement polynomila of the second kind associated with Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.

math.NT↗

arXiv2024

Probabilistic proof of a summation formula

The aim of this paper is to derive a summation formula for the alternating infinite series and an expression for zeta function by using hyperbolic secant random variables. These identities involve Euler numbers and are obtained by computing the moments of the random variable and the moments of the sum of two independent such random variables.

math.NT↗

arXiv2024

Probabilistic degenerate Bernstein polynomials

In recent years, both degenerate versions and probabilistic extensions of many special numbers and polynomials have been explored. For instance, degenerate Bernstein polynomials and probabilistic Bernstein polynomials were investigated earlier. Assume that Y is a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study probabilistic degenerate Bernstein polynomials associated with Y which are both probabilistic extension of the degenerate Bernstein polynomials and degenerate version of the probabilistic Bernstein polynomials associated with $Y$. We derive several explicit expressions and certain related identities for those polynomials. In addition, we treat the special cases of the Poisson random variable, the Bernoulli random variable and of the binomial random variable.

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