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 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 Higher order bell polynomials and the relevant integer sequences

2017 ◽  
Vol 11 (2) ◽  
pp. 327-339 ◽  
Author(s):  
Pierpaolo Natalini ◽  
Paolo Ricci
Keyword(s):  
Recurrence Relation ◽  
Computer Algebra ◽  
Higher Order ◽  
Bell Polynomials ◽  
Integer Sequences ◽  
Composite Function

The recurrence relation for the coefficients of higher order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a multiple composite function, is proved. Therefore, starting from this recurrence relation and by using the computer algebra program Mathematica?, some tables for complete higher order Bell polynomials and the relevant numbers are derived.

scholarly journals Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

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 Recurrence Relations for Strongly q-Log-Convex Polynomials

2011 ◽  
Vol 54 (2) ◽  
pp. 217-229 ◽  
Author(s):  
William Y. C. Chen ◽  
Larry X. W. Wang ◽  
Arthur L. B. Yang
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Bell Polynomials ◽  
Bessel Polynomials

AbstractWe consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.

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 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 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>

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