Moment Functions on Groups

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

Some extensions for the several combinatorial identities

In this paper, we give some extensions for Mortenson’s identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving the harmonic numbers.

Dynamics of abundant solutions to the generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

The present paper is devoted to discussion of (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation from point of view of their multi-soliton solutions and localized solutions associated with multi-soliton solutions. Firstly, taking advantage of the Bell-polynomial approach, we construct the Hirota bilinear form of (1.1). Based on that, the multi-soliton solutions are also singled out. Subsequently, the (3+1)-dimensional gBKP equation is also found to allow fruitful localized solutions, including breather, lump, rogue wave, and hybrid solutions. These results obtained in this work adequately illustrate the effectiveness of the long wave limit method and complex conjugate technique, which are expected to be employed to obtain more abundant exact solutions.

The integrability of the coupled Ramani equation with binary Bell polynomials

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

Generalized (3+1)-dimensional Boussinesq equation: Breathers, rogue waves and their dynamics

In this work, we consider the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq equation, which can describe the propagation of gravity wave on the surface of water. Based on the Bell polynomial theory, a powerful technique is employed to explicitly construct its bilinear formalism and two-soliton solutions, based on which the new rational solution is well-constructed. Moreover, the extended homoclinic test approach is presented to succinctly construct the breather wave and rogue wave solutions of the Boussinesq equation. Then the main characteristics of these solutions are graphically discussed. More importantly, they reveal that the extreme behavior of the breather wave can give rise to the rogue wave.

Characteristics of the breather waves, lump waves and semi-rational solutions in a generalized (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

In this work, we study a generalized (2[Formula: see text]+[Formula: see text]1)-dimensional asymmetrical Nizhnik–Novikov–Veselov (NNV) equation. Its Hirota bilinear form is constructed via the Bell polynomial. Based on the obtained bilinear form, the Nth-order breather waves are derived explicitly under certain parameter constraints. Moreover, we generate the nonsingular Nth-order lump waves through applying the long wave limit method. Additionally, we successfully present the semi-rational waves containing the combination of lump waves and single-soliton waves, the combination of lump waves and breather waves.