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

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.

scholarly journals Some identities of Lah–Bell polynomials

2020 ◽  
Vol 2020 (1) ◽  
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.

scholarly journals Fourier series of functions involving higher-order ordered Bell polynomials

2017 ◽  
Vol 15 (1) ◽  
pp. 1606-1617 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Gwan-Woo Jang ◽  
Lee Chae Jang
Keyword(s):  
Number Theory ◽  
Fourier Series ◽  
Higher Order ◽  
Series Expansions ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Enumerative Combinatorics ◽  
Bernoulli Functions ◽  
Series Of Functions

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.

scholarly journals On Generalized Bell Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Roberto B. Corcino ◽  
Cristina B. Corcino
Keyword(s):  
Recurrence Relation ◽  
Bell Polynomials ◽  
Bell Numbers

It is shown that the sequence of the generalized Bell polynomialsSn(x)is convex under some restrictions of the parameters involved. A kind of recurrence relation forSn(x)is established, and some numbers related to the generalized Bell numbers and their properties are investigated.

scholarly journals Bell Numbers Modulo a Prime Number, Traces and Trinomials

10.37236/3532 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Luis H. Gallardo ◽  
Olivier Rahavandrainy
Keyword(s):  
Prime Number ◽  
Finite Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bell Numbers ◽  
Fixed Integer ◽  
Bell Number ◽  
Modulo P ◽  
Necessary And Sufficient

Given a prime number $p$, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition $\beta(n) =k\beta(0)$ in $\mathbb{F}_{p^p}$ in order to resolve the congruence $B(n) \equiv k \pmod{p}$, where $B(n)$ is the $n$-th Bell number, and $k$ is any fixed integer. Several applications of the formula and of the condition are included, in particular we give equivalent forms of the conjecture of Kurepa that $B(p-1)$ is $\neq 1$ modulo $p$.

scholarly journals Generalized Bell Numbers and Peirce Matrix via Pascal Matrix

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Eunmi Choi
Keyword(s):  
Bell Numbers ◽  
Pascal Matrix ◽  
Bell Number

With the Stirling matrix S and the Pascal matrix T, we show that TkS (k≥0) satisfies a type of generalized Stirling recurrence. Then, by expressing the sum of components of each row of TkS as k-Bell number, we investigate properties of k-Bell numbers as well as k-Peirce matrix.

A new set of Sheffer–Bell polynomials and logarithmic numbers

2019 ◽  
Vol 26 (3) ◽  
pp. 367-379 ◽  
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.

scholarly journals A new approach to Bell and poly-Bell numbers and polynomials

2021 ◽  
Vol 7 (3) ◽  
pp. 4004-4016
Author(s):  
Taekyun Kim ◽  
◽  
Dae San Kim ◽  
Dmitry V. Dolgy ◽  
Hye Kyung Kim ◽  
...  
Keyword(s):  
Recurrence Relations ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
New Approach ◽  
Special Polynomials ◽  
Explicit Expressions

<abstract><p>Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.</p></abstract>

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

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>

scholarly journals The r-Dowling Numbers and Matrices Containing r-Whitney Numbers of the Second Kind and Lah Numbers

2019 ◽  
Vol 12 (3) ◽  
pp. 1122-1137
Author(s):  
Roberto Bagsarsa Corcino ◽  
Charles Montero ◽  
Maribeth Montero ◽  
Jay Ontolan
Keyword(s):  
Explicit Formula ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Whitney Numbers

This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms  of the $r$-Whitney numbers of the second kind and the ordinary Lah numbers. As a consequence, a relation between $(r,\beta)$-Bell numbers and the sums of row entries of the product of two matrices containing the $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established.  Moreover, a $q$-analogue of the explicit formula is obtained.

scholarly journals The Kurepa-Vandermonde matrices arising from Kurepa’s left factorial hypothesis

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2207-2215
Author(s):  
Romeo Mestrovic
Keyword(s):  
Greatest Common Divisor ◽  
Bell Numbers ◽  
Vandermonde Matrices ◽  
Bell Number ◽  
Mod P

Kurepa?s (left factorial) hypothesis asserts that for each integer n ? 2 the greatest common divisor of !n := Pn-1?k=0 k! and n! is 2. It is known that Kurepa?s hypothesis is equivalent to p-1?k=0 (-1)k/k!?/ 0 (mod p) for each odd prime p, or equivalently, Sp-1 ?/ 0(modp) (i.e., Bp-1 ?/1(modp)) for each odd prime p, where Sp-1 and Bp-1 are the (p-1)th derangement number and the (p-1)th Bell number, respectively. Motivated by these two reformulations of Kurepa?s hypothesis and a congruence involving the Bell numbers and the derangement numbers established by Z.-W. Sun and D. Zagier [28, Theorem 1.1], here we give two ?matrix? formulations of Kurepa?s hypothesis over the field Fp, where p is any odd prime. The matrices Vp and Cp which are involved in these ?matrix? formulations of Kurepa?s hypothesis are the square (p-1)x(p-1) Vandermondelike matrices. Accordingly, Vp and Cp are called the Kurepa-Vandermonde matrices. Furthermore, for each odd prime p we determine det(Vp) and det(Cp) in the field Fp.

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