scholarly journals Closed formulas for special bell polynomials by Stirling numbers and associate Stirling numbers

2020 ◽  
Vol 108 (122) ◽  
pp. 131-136
Author(s):  
Feng Qi ◽  
Dongkyu Lim
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Special Values

We derive two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of associate Stirling numbers of the second kind, give an explicit formula for associate Stirling numbers of the second kind in terms of the Stirling numbers of the second kind, and, consequently, present two explicit formulas for two sequences of special values of the Bell polynomials of the second kind in terms of the Stirling numbers of the second kind.

scholarly journals On r-Central Incomplete and Complete Bell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 724 ◽  
Author(s):  
Dae San Kim ◽  
Han Young Kim ◽  
Dojin Kim ◽  
Taekyun Kim
Keyword(s):  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Central Factorial Numbers

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well.

scholarly journals Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Stirling Numbers ◽  
Higher Order ◽  
Recursive Relation ◽  
Order Derivative ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Derivative Formula ◽  
Higher Order Derivative ◽  
Determinantal Expression

In the paper, using two inversion theorems for the Stirling numbers and binomial coecients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two dierentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

scholarly journals Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 319-327 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Integral Representations ◽  
Logarithmic Function ◽  
Explicit Formulas

In the paper, by establishing a new and explicit formula for computing the n-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial n! are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.

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 A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers

2015 ◽  
Vol 3 (1) ◽  
pp. 33 ◽  
Author(s):  
Bai-Ni Guo ◽  
Feng Qi
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Genocchi Numbers

<p>In the paper, the authors concisely review some explicit formulas and establish a new explicit formula for the Bernoulli and Genocchi numbers in terms of theStirlingnumbers of the second kind.</p>

scholarly journals An explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Explicit Formula ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Explicit Formulas

We derive an explicit formula for the Bernoulli numbers in terms of Stirling numbers of the second kind. To the best of our knowledge, the formula is new. Additionally, we recover two more explicit formulas from this formula expressing the Bernoulli numbers in terms of Stirling numbers of the second kind.

scholarly journals Some Inequalities of the Bell Polynomials

2017 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Explicit Formula ◽  
Generating Functions ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Higher Order ◽  
Complete Monotonicity ◽  
Bell Polynomials ◽  
Logarithmic Convexity ◽  
Inversion Theorem ◽  
Derivatives Of

In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Fa&agrave; di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.

scholarly journals Generalizations of the Bell Numbers and Polynomials and Their Properties

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Dongkyu Lim ◽  
Bai-Ni Guo
Keyword(s):  
Stirling Numbers ◽  
Bell Polynomials ◽  
Bell Numbers ◽  
Explicit Formulas ◽  
Logarithmic Convexity ◽  
Inversion Formulas ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions ◽  
And Inversion

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Fa&agrave; di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

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