Degenerate Poly‐Lah‐Bell Polynomials and Numbers

Kim, Taekyun; Kim, Hye Kyung

doi:10.1155/2022/2917943

Cited by 8 publications

(9 citation statements)

References 9 publications

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“…In recent years, there have been active explorations for various degenerate versions of many special numbers and polynomials with diverse tools such as generating functions, combinatorial methods, p-adic analysis, umbral calculus, differential equations, probability theory, operator theory, analytic number theory and quantum physics. These were initiated by Carlitz in [3,4] and yielded many interesting results of arithmetical and combinatorial nature (see [7][8][9][12][13][14][16][17][18] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

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Dimorphic Mersenne numbers and their applications

Kim¹,

Ким²

2022

Preprint

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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.M n = 2 n − 1, (n ≥ 0), (see [2]).

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Section: Introductionmentioning

confidence: 99%

“…, where B n are the ordinary Bernoulli numbers, (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). From ( 4), we note that [3,4]).…”

Section: Introductionmentioning

confidence: 99%

Dimorphic Mersenne numbers and their applications

Kim¹,

Ким²

2022

Preprint

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“…The unsigned Lah numbers L(n, k) counts the number of ways of a set of n elements can be partitioned into k nonempty linearly ordered subsets. These numbers are generalized to the multi-Lah numbers L (k 1 ,k 2 ,...,k r ) (n, r) (see (8)) which boil down to the unsigned Lah numbers for (k 1 , k 2 , . .…”

Section: Introductionmentioning

confidence: 99%

“…We recall that the falling factorial sequence is given by [1][2][3][4][5][7][8][9][10][11][12][13][14][15][16]18,19]). (1) It is well known that the Stirling numbers of the second kind are defined by…”

Section: Introductionmentioning

confidence: 99%

Multi-Lah numbers and multi-Stirling numbers of the first kind

Ким

Kim

et al. 2021

Adv Differ Equ

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In this paper, we introduce multi-Lah numbers and multi-Stirling numbers of the first kind and recall multi-Bernoulli numbers, all of whose generating functions are given with the help of multiple logarithm. The aim of this paper is to study several relations among those three kinds of numbers. In more detail, we represent the multi-Bernoulli numbers in terms of the multi-Stirling numbers of the first kind and vice versa, and the multi-Lah numbers in terms of multi-Stirling numbers. In addition, we deduce a recurrence relation for multi-Lah numbers.

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“…k! are the (unsigned) Lah numbers (see earlier studies [2,10,24,30]) whose generating function is given by…”

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confidence: 99%

Central factorial numbers associated with sequences of polynomials

Ким

Kim

2023

Math Methods in App Sciences

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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 P = {p n (x)} ∞ n=0 such that deg p n (x) = n, p 0 (x) = 1. The aim of this paper is to study the central factorial numbers of the second kind associated with any sequence of polynomials and of the first kind associated with any sequence of polynomials, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with any sequence of polynomials 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 any sequence of polynomials to each of our examples.

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Degenerate Poly‐Lah‐Bell Polynomials and Numbers

Cited by 8 publications

References 9 publications

Dimorphic Mersenne numbers and their applications

Dimorphic Mersenne numbers and their applications

Multi-Lah numbers and multi-Stirling numbers of the first kind

Central factorial numbers associated with sequences of polynomials

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