Partial Bell Polynomials Research Articles

Several Derivative Formulas of Two Exponential Functions and Real Power of Hyperbolic Secant Function with a Generalization of a Formula for Specific Partial Bell Polynomials

In the paper, by virtue of some identities for the partial Bell polynomials and with the aid of the Faá di Bruno formula, the author presents several derivative formulas of two exponential functions and the real power of the hyperbolic secant function, and generalizes a formula for specific partial Bell polynomials.

Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant π and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

Series expansions for powers of sinc function and closed-form expressions for specific partial bell polynomials

In the paper, with the aid of the Fa? di Bruno formula, in terms of the central factorial numbers and the Stirling numbers of the second kinds, the authors derive several series expansions for any positive integer powers of the sinc and sinhc functions, discover several closed-form expressions for partial Bell polynomials of all derivatives of the sinc function, establish several series expansions for any real powers of the sinc and sinhc functions, and present several identities for central factorial numbers of the second kind and for the Stirling numbers of the second kind.

The 2-successive partial Bell polynomials

In this paper, we discuss a new class of partial Bell polynomials. The first section gives an overview of partial Bell polynomials and their related 2-successive Stirling numbers. In the second section, we introduce the concept of 2-successive partial Bell polynomials. We give an explicit formula for computing these polynomials and establish their generating function. In addition, we derive several recurrence relations that govern the behaviour of these polynomials. Furthermore, we study specific cases to illustrate the applicability and versatility of this new class of polynomials.

A Sufficient and Necessary Condition for the Power-Exponential Function 1+1xαx to Be a Bernstein Function and Related nth Derivatives

In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1xαx and (1+x)α/x, and present a closed-form formula of the partial Bell polynomials Bn,k(H0(x),H1(x),⋯,Hn−k(x)) for n≥k≥0, where Hk(x)=∫0∞eu−1−ueuuk−1e−xudu for k≥0 are completely monotonic on (0,∞).

Large-scale behaviour and hydrodynamic limit of beta coalescents

We quantify the behaviour at large scales of the beta coalescent Π={Π(t),t≥0} with parameters a,b>0. Specifically, we study the rescaled block size spectrum of Π(t) and of its restriction Πn(t) to {1,…,n}. Our main result is a law of large numbers type of result if Π comes down from infinity. In the case of Kingman’s coalescent the derivation of this so-called hydrodynamic limit has been known since the work of Smoluchowski (Z. Phys. 17 (1916) 557–585). We extend Smoluchowski’s result to beta coalescents and show that if Π comes down from infinity both rescaled spectra n−1(c1Π(tτn),…,cnΠ(tτn)),andn−1(c1Πn(tτn),…,cnΠn(tτn)), converge to (different) deterministic limits that we compute explicitly in terms of partial Bell polynomials. Here ciπ counts the number of blocks of size i in a partition π, and (τn) is a sequence such that τn∼n−(1−a) as n→∞. Along the way we study the nontrivial limits of the rescaled block counting processes {nα#Πn(tτn),t≥0}, and {nα#Π(tτn),t≥0}, where α∈[−1,−2/(3−a)), and τn∼nα(1−a) if Π comes down from infinity.

Closed-Form Formulas for the nth Derivative of the Power-Exponential Function xx

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function xx, establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function xx and for related functions and integer sequences.

A Relationship between the Partial Bell Polynomials and Alternating Run Polynomials

A Relationship between the Partial Bell Polynomials and Alternating Run Polynomials

Specific values of partial bell polynomials and series expansions for real powers of functions and for composite functions

Starting from Maclaurin?s series expansions for positive integer powers of analytic functions, the authors derive an explicit formula for specific values of partial Bell polynomials, present a general term of Maclaurin?s series expansions for real powers of analytic functions, obtain Maclaurin?s series expansions of some composite functions, recover Maclaurin?s series expansions for real powers of inverse sine function and sinc function, recover a combinatorial identity involving the falling factorials and the Stirling numbers of the second kind, deduce an explicit formula of the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, recover an explicit formula of the Bernoulli numbers in terms of the Stirling numbers of the second kind, recover an explicit formula of the Bell numbers in terms of the Stirling numbers of the second kind, reformulate three specific partial Bell polynomials in terms of central factorial numbers of the second kind, and present some Maclaurin?s series expansions and identities related to the Euler numbers and their generating function.

IDENTITIES INVOLVING GENERALIZED BERNOULLI NUMBERS AND PARTIAL BELL POLYNOMIALS WITH THEIR APPLICATIONS

In the present paper, we show that the partial Bell polynomials allow for obtaining identities involving the generalized Bernoulli numbers. Then, on applying these identities we derive different generating and bilateral generating functions.

Several recursive and closed-form formulas for some specific values of partial Bell polynomials

In this paper, the authors derive several recursive and closed-form formulas for some specific values of partial Bell polynomials.

Solution of the equation $y^{\prime}=f(y)$ and Bell Polynomials

In this paper we use Faa di Bruno's formula to associate Bell polynomial values to differential equations of the form $y^{\prime}=f(y)$. That is, we use partial Bell polynomials to represent the solution of such an equation and use the solution to compute special values of partial Bell polynomials.

Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi

Abstract In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.

Degenerate Fubini-Type Polynomials and Numbers, Degenerate Apostol–Bernoulli Polynomials and Numbers, and Degenerate Apostol–Euler Polynomials and Numbers

In this paper, by introducing degenerate Fubini-type polynomials, with the help of the Faà di Bruno formula and some properties of partial Bell polynomials, the authors provide several new explicit formulas and recurrence relations for Fubini-type polynomials and numbers, associate the newly defined degenerate Fubini-type polynomials with degenerate Apostol–Bernoulli polynomials and degenerate Apostol–Euler polynomials of order α. These results enable one to present additional relations for some degenerate special polynomials and numbers.

Applications of the differential transformation to three-point singular boundary value problems for ordinary differential equations

The differential transform method is used to find numerical approximations of the solution to a class of certain nonlinear three-point singular boundary value problems. The method is based on Taylor's theorem. Coefficients of the Taylor series are determined by constructing a recurrence relation. To deal with the nonlinearity of the problems, the Faa di Bruno's formula containing the partial ordinary Bell polynomials is applied within the differential transform. The error estimation results are also presented. Four concrete problems are studied to show efficiency and reliability of the method. The obtained results are compared to other methods, e.g., reproducing kernel Hilbert space method.

Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important, useful, and applicable in combinatorial number theory.

MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

Some combinatorial interpretation related to partial Bell polynomials

In this paper, using a combinatorial approach, we give a combinatorial interpretation of sequences of numbers related to the partial Bell polynomials. Some recurrence relations for these polynomials may be deduced.

Bell Polynomials and Nonlinear Inverse Relations

By means of the Lagrange expansion formula, we establish a general pair of nonlinear inverse series relations, which are expressed via partial Bell polynomials with the connection coefficients involve an arbitrary formal power series. As applications, two examples are presented with one of them recovering the difficult theorems discovered recently by Birmajer, Gil and Weiner (2012 and 2019).

Some properties of degenerate complete and partial Bell polynomials

In this paper, we study degenerate complete and partial Bell polynomials and establish some new identities for those polynomials. In addition, we investigate the connections between modified degenerate complete and partial Bell polynomials, which are closely related to the degenerate complete and partial Bell polynomials, and the joint distribution of weighted sums of independent degenerate Poisson random variables.