A Relationship between the Partial Bell Polynomials and Alternating Run Polynomials

In this note, we first derive an exponential generating function of the alternating run polynomials. We then deduce an explicit formula of the alternating run polynomials in terms of the partial Bell polynomials.

Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized NizhnikNovikov-Veselov (GNNV) equation.The bilinear representation,bilinear B&cklund transformation,Lax pair and infiniteconservation laws of the GNNV equation are obtained directly,without too much trick like Hirota’s bilinear method.

A Kind of Identities Involving Complete Bell Polynomials

The purpose of this paper is to establish a formula of higher derivative by Faà di Bruno formula, and apply it to some known results to get some identities involving complete Bell polynomials.

Integrability of an extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended （2＋ 1）-dimensional shaUow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Backlund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.

On Polynomials Rn(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on

In this paper, one introduces the polynomials Rn(x) and numbers Rn and derives some interesting identities related to the numbers and polynomials: Rn and Rn(x). We also give relation between the Stirling numbers, the Bell numbers, the Rn and Rn(x).

The Modified Kadomtsev-Petviashvili Equation with Binary Bell Polynomials

Binary Bell Polynomials play an important role in the characterization of bilinear equation. The bilinear form, bilinear B?cklund transformation and Lax pairs for the modified Kadomtsev-Petviashvili equation are derived from the Binary Bell Polynomials.

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of(2+l)-Dimensional Nonlinear Evolution Equations

In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.

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.

A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given.

Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (Ⅱ)

In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.

Bilinear forms through the binary Bell polynomials, N solitons and Backlund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.

Trilinear equations, Bell polynomials, and resonant solutions

A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative examples are made by an algorithm using weights of dependent variables.

Bell Polynomial Approach for the Solutions of Fredholm Integro- Differential Equations with Variable Coefficients

In this article,we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials.Using collocation points and treating the solution as a linear combination of Bell polynomials,the problem is reduced to linear system of equations whose unknown variables are Bell coefficients.The solution to this algebraic system determines the approximate solution.Error estimation of approximate solution is done.Some examples are provided to illustrate the performance of the method.The numerical results are compared with the collocation method based on Legendre polynomials and the other two methods based on Taylor polynomials.It is observed that the method is better than Legendre collocation method and as accurate as the methods involving Taylor polynomials.

Binary Bell polynomial application in generalized(2+1)-dimensional KdV equation with variable coefficients

In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized （2＋1）-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Backlund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.

Bell Polynomial Approach and N-Soliton Solutions for a Coupled KdV-mKdV System

In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.

On Integrable Properties for Two Variable-Coefficient Evolution Equations

With the help of the extended binary Bell polynomials, the new bilinear representations, Backlund trans- formations, Lax pair and infinite conservation laws for two types of variable-coefficient nonlinear integrable equations are obtained, respectively, which are more straightforward than previous corresponding results obtained. Finally, we obtain new multi-soliton wave solutions of a reduced soliton equations with variable coefficients.

Expansions for the Distribution and the Maximum from Distributions with an Asymptotically Gamma Tail When a Trend Is Present

We give expansions about the Gumbel distribution in inverse powers of n and logn for Mn, the maximum of a sample size n or n ＋ 1 when the j-th observation is μ（j/n） ＋ ej, μ is any smooth trend flmction and the residuals {ej} are independent and identically distributed with P（e〉r）≈exp（-δx）x^do ∑k=1 ∞ckx^-kβ as -x→∞. We illustrate practical value of the expansions using simulated data sets.

Exact Periodic Wave Solution of Extended(2+1)-Dimensional Shallow Water Wave Equation with Generalized D_-operators

With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dpˉ-operators in term of the Hirota direct method when the appropriate value of pˉ is determined. Furthermore, the resulting approach is applied to solve the extended(2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.

ON THE DIVIDED DIFFERENCE FORM OF FAA DI BRUNO＇S FORMULA

The n-divided difference of the composite function h ：= f o g of functions f, g at a group of nodes t0,t1,…,tn is shown by the combinations of divided differences of f at the group of nodes g（t0）,g（t1）,…,g（tm） and divided differences of g at several partial group of nodes t0,t1,…,tn, where m = 1,2,…,n. Especially, when the given group of nodes are equal to each other completely, it will lead to Faà di Bruno＇s formula of higher derivatives of function h.

Bcklund Transformations and Solutions of a Generalized Kadomtsev-Petviashvili Equation

In this paper, the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials. The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function, respectively. And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution. In the end, the bilinear Backlund transformations are derived.