Fourier series of functions involving higher-order ordered Bell polynomials

AbstractIn 1859, Cayley introduced the ordered Bell numbers which have been used in many problems in number theory and enumerative combinatorics. The ordered Bell polynomials were defined as a natural companion to the ordered Bell numbers (also known as the preferred arrangement numbers). In this paper, we study Fourier series of functions related to higher-order ordered Bell polynomials and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.

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Fourier series of functions associated with higher-order Bernoulli polynomials

The Journal of Nonlinear Sciences and Applications ◽

10.22436/jnsa.010.05.25 ◽

2017 ◽

Vol 10 (05) ◽

pp. 2579-2591

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Dmitry V. Dolgy ◽

Jin-Woo Park

Keyword(s):

Fourier Series ◽

Bernoulli Polynomials ◽

Higher Order ◽

Series Of Functions

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Fourier series of higher-order Bernoulli functions and their applications

Journal of Inequalities and Applications ◽

10.1186/s13660-016-1282-y ◽

2017 ◽

Vol 2017 (1) ◽

Cited By ~ 2

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Seog-Hoon Rim ◽

Dmitry V. Dolgy

Keyword(s):

Fourier Series ◽

Higher Order ◽

Bernoulli Functions

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A new set of Sheffer–Bell polynomials and logarithmic numbers

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2007 ◽

2019 ◽

Vol 26 (3) ◽

pp. 367-379 ◽

Cited By ~ 3

Author(s):

Gabriella Bretti ◽

Pierpaolo Natalini ◽

Paolo Emilio Ricci

Keyword(s):

Higher Order ◽

Bell Polynomials ◽

Bell Numbers ◽

Exponential Polynomials ◽

Polynomial Sequences

Abstract In a recent paper, we have introduced new sets of Sheffer and Brenke polynomial sequences based on higher order Bell numbers. In this paper, by using a more compact notation, we show another family of exponential polynomials belonging to the Sheffer class, called, for shortness, Sheffer–Bell polynomials. Furthermore, we introduce a set of logarithmic numbers, which are the counterpart of Bell numbers and their extensions.

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Some identities of Lah–Bell polynomials

Advances in Difference Equations ◽

10.1186/s13662-020-02966-6 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Yuankui Ma ◽

Dae San Kim ◽

Taekyun Kim ◽

Hanyoung Kim ◽

Hyunseok Lee

Keyword(s):

Nonnegative Integer ◽

Bernoulli Polynomials ◽

Higher Order ◽

Umbral Calculus ◽

Bell Polynomials ◽

Bell Numbers ◽

Bell Number ◽

Natural Extensions

Abstract Recently, the nth Lah–Bell number was defined as the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets for any nonnegative integer n. Further, as natural extensions of the Lah–Bell numbers, Lah–Bell polynomials are defined. We study Lah–Bell polynomials with and without the help of umbral calculus. Notably, we use three different formulas in order to express various known families of polynomials such as higher-order Bernoulli polynomials and poly-Bernoulli polynomials in terms of the Lah–Bell polynomials. In addition, we obtain several properties of Lah–Bell polynomials.

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Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers

AIMS Mathematics ◽

10.3934/math.2022161 ◽

2022 ◽

Vol 7 (2) ◽

pp. 2929-2939

Author(s):

Hye Kyung Kim ◽

Keyword(s):

Generating Functions ◽

Stirling Numbers ◽

Bell Polynomials ◽

Bell Numbers ◽

Explicit Formulas ◽

Riemann Integral ◽

Enumerative Combinatorics ◽

Central Factorial Numbers ◽

Number Counts

<abstract><p>The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.</p></abstract>

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Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

Advances in Difference Equations ◽

10.1186/s13662-021-03258-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Lee-Chae Jang ◽

Hyunseok Lee ◽

Han-Young Kim

Keyword(s):

Bell Polynomials ◽

Bell Numbers ◽

Bell Number ◽

Finite Sums

AbstractThe nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

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