Generating Function Method Research Articles

Applying the monomiality principle to the new family of Apostol Hermite Bernoulli-type polynomials

Abstract In this article, we introduce a new class of polynomials, known as Apostol Hermite Bernoulli-type polynomials, and explore some of their algebraic properties, including summation formulas and their determinant form. The majority of our results are proven using generating function methods. Additionally, we investigate the monomiality principle related to these polynomials and identify the corresponding derivative and multiplicative operators.

Further Results for Hermite-Based Milne-Thomson Type Fubini Polynomials with Trigonometric Functions

This paper examines generating functions of r-parametric Hermite-based Milne-Thomson polynomials. Using generating function methods, the relationships among these polynomials, Fubini type polynomials, and trigonometric functions are given. Moreover, new formulas are derived by utilizing not only the generating functions of these polynomials but also associated functional equations. These formulas pertain to r-parametric Hermite-based sine-and cosine-Milne-Thomson Fubini polynomials, as well as Stirling type polynomials and numbers. Additionally, by analyzing special cases of newly obtained results, some known formulas are also derived. Furthermore, some identities involving secant and cosecant numbers are derived through the properties of trigonometric functions. Special polynomials and their generating functions are an important tool for solving some problems in many areas such as combinatorics and number theory. By introducing new formulas, this paper significantly enhances these problems-solving abilities in these areas. Consequently, these results have potential to shed light on important applications in mathematics, engineering, and mathematical physics.

Properties of the connected components in projections of random bipartite networks: effects of clique size fluctuations

Projection is a helpful description for treating bipartite networks as (monopartite) networks with pairwise interactions. Projections induce correlation spontaneously, avoiding negative degree correlation, even if bipartite networks are entirely random. In this study, we examined the structure of projections of random bipartite networks characterized by the degree distribution of individual and group nodes through the generating function method. We decomposed a projection into two subgraphs, the giant component, and finite components and analyzed their degree correlation. We demonstrate that positive degree correlations in projections originating from the clique size fluctuation remain after the decomposition at the set of finite components, although the values of their clustering coefficient are still finite. The giant component can exhibit either positive or negative degree correlations based on the structure of the projection. However, they are positively correlated in most cases. In addition, we found that a projection removed the giant component coincides with one in the subcritical phase, i.e., the discrete duality relation, when the degree distributions for group and individual are of Poisson.

Partial Derivative Equations and Identities for Hermite-Based Peters-Type Simsek Polynomials and Their Applications

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials.

Some Identities on λ-Analogues of Lah Numbers and Lah-Bell Polynomials

In recent years, some applications of Lah numbers were discovered in the real world problem of telecommunications and optics. The aim of this paper is to study the λ-analogues of Lah numbers and Lah-Bell polynomials which are λ-analogues of the Lah numbers and and Lah-Bell polynomials. Here we note that λ-analogues appear when we replace the falling factorials by the generalized falling factorials in the defining equations. By using generating function method, we study some properties, explicit expressions, generating functions and Dobinski-like formulas for those numbers and polynomials. We also treat the more general λ-analogues of r-Lah numbers and r-extended λ-Lah-Bell polynomials. In addition, we show that the expectations of two random variables, both associated with the Poisson random variable with parameter α , are equal to the λ-analogue of the Lah-Bell polynomial evaluated at α for one and the r-extended λ-Lah-Bell polynomial evaluated at α for the other.

Exact analysis of generalized degree-based percolation without memory

We study the problem of generalized degree-based percolation without memory, where the probability of node removal depends on a non-negative function of node degree within the remaining network. We derive a set of nonlinear ordinary differential equations describing the evolution of the degree distribution during the percolation process. Employing generating function methods, we calculate various quantities of interest such as the probability of a randomly selected node belonging to the giant component, the mean cluster size, and the critical point at which a network transitions from connected to fragmented. Validation of our approach is achieved through extensive Monte Carlo simulations and numerical solutions. Our results indicate that removing highly connected nodes with small preference probabilities significantly lowers the critical point compared to random removal. Moreover, our analytical framework is applicable to random networks generated by the configuration model with any bounded degree distribution. It provides a comprehensive and systematic approach to analyze network robustness against various degree-based attack strategies, offering valuable insights into network design and protection.

Analysis of the chemical diffusion master equation for creation and mutual annihilation reactions

We propose an infinite dimensional generating function method for finding the analytical solution of the so-called chemical diffusion master equation (CDME) for creation and mutual annihilation chemical reactions. CDMEs model by means of an infinite system of coupled Fokker–Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Using our method we are able to link certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space. Our technique extends the one presented in Lanconelli [J. Math. Anal. Appl. 526, 127352 (2023)] and Lanconelli et al. [arXiv:2302.10700 [math.PR] (2023)] which allowed for an explicit representation for the solution of birth-death type CDMEs.

Combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers

<abstract><p>In this work, by means of the generating function method and the De Moivre's formula, we derive some interesting combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers. One of them confirms the formula proposed recently by Svinin (2022).</p></abstract>

Monophonic Polynomial of the cartesian product of some graphs

Let be the family of monophonic sets of a graph with cardinality and let Then the monophonic polynomial of is defined as , where is the monophonic number of . In this paper we have determined the sufficient condition for the monophonic set of . Also, we have calculated the monophonic polynomial of the cartesian product of some specific graphs by generating function method.

Analysis of M/M/1 queueing systems with negative customers and unreliable repairers

In this article, we study the M/M/1 queueing systems with negative customers and unreliable repairers. The arrival of a negative customer causes the failure of the server, and the server is repaired immediately. The repairers may successfully repair the server with probability p, and the server continues to serve customers in the system. Otherwise, once the repair fails, the system will be cleared and all customers are forced to leave the system. By analyzing the quasi-birth-and-death process and using the probability generating function method, we obtain the steady-state probability and some performance measures of the system. Then, based on the reward-cost structure, we discuss the strategic behavior of customers and consider the equilibrium strategy. Finally, the effects of system parameters on the performance measures, individual benefit, and social benefit are analyzed by numerical examples.

Key Factors Affecting the Capacity Value of New Energy and Sensitivity Analysis

The capacity credit (CC) of new energy describes the contribution of new energy to capacity adequacy and power system reliability. Many factors affect the CC of renewable energy systems. It is necessary to study various measures to improve the capacity credit of new energy. In this paper, the CC assessment model of a (photovoltaic) PV power plant based on the effective load carrying capability (ELCC) method is established by the universal generating function method. In addition, the sensitivity analysis of eight different key factors affecting the CC of photovoltaic power plants, such as PV penetration, number of PV output states, PV module failure characteristics, and other eight key factors, is carried out to point out how various factors affect the CC of the new energy. Analyzing the degree and depth of influence of key factors on new energy CC, which can provide reference for new energy CC.

Reliability assessment of performance-based balanced systems with rebalancing mechanisms

Performance-based balanced systems (PBSs) are widely used as critical equipment in many fields, so their reliability assessment has become a hot topic. Taking the battery pack PBS as an example, the balance means the performance differences between components are within an acceptable range, where the common bus is used to redistribute the performance for rebalancing, and the redistribution is affected by the transmission loss and transmission capacity limit. To reflect the above rebalancing mechanisms, a reliability evaluation method is proposed for PBSs with common bus performance sharing (PBSs-CBPS) considering balance degree threshold, transmission loss, and transmission capacity limit. Firstly, a continuous-time discrete-state Markov model is built to address the transition behaviors of components. Next, considering the effects of balance degree threshold and transmission loss, the rebalancing process is formulated as a nonlinear programming problem to obtain the performance of components after rebalancing, and the system reliability model is established. Then, the universal generation function (UGF) method combined with nonlinear programming is proposed to calculate system reliability, and Paper-Cut UGF Algorithm (PCUGF) is proposed to improve computational efficiency. Finally, an analytical example and a numerical example of 6S1P lithium-ion battery packs are given to show the effectiveness of the proposed method.

Strategic Analysis of Retrial Queues with Setup Times, Breakdown and Repairs

This paper considers a repairable M/M/1 retrial queueing model with setup times. Once the system is empty, the server will be closed down to reduce operating costs. And the system will be activated only when a new customer arrives. The customer who activates the server will enter the retrial orbit waiting to reapply for service. The server may break down during the busy period. First, the steady-state probability of the model is obtained by using the probability generating function method. And we derive performance measures of the system such as the queue length of the orbit, the numerical examples are given to show the sensitivity of the performance measures. Second, the cost function is established to find the minimum cost of the system, and we study the effects of some parameters on the cost by numerical examples. Finally, from the perspective of the customer and social planner, we construct the individual utility function and the social welfare function in the almost and fully unobservable cases, and then the optimal strategy of the customers is analyzed.

On Apostol-Type Hermite Degenerated Polynomials

This article presents a generalization of new classes of degenerated Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol–Bernoulli polynomials. These results are shown using generating function methods for Apostol–Euler and Apostol–Genocchi Hermite polynomials of level m.

Generalized Mixed Type Bernoulli-Gegenbauer Polynomials

The generalized mixed type Bernoulli-Gegenbauer polynomials of order α >−1/2 are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. The main purpose of this paper is to discuss some of their algebraic and analytic properties.

Specification transformation method for functional program generation based on partition-recursion refinement rule

Implementations that follow the functional programming paradigm are being used in more and more domains. As functional programming paradigm has mathematical reference transparency, refinement to functional programs contributes to improving the reliability of the transformation process and simplifying the refinement steps. However, it is a challenge to generate functional programs from specifications. Most existing transformation methods refine specifications into abstract algorithm-level programs based on loop invariants rather than functional programs. This paper proposes a novel functional program generation method based on the partition-recursion refinement rule. It establishes a novel program refinement framework based on functional theory for the first time. This is the first study to regard the whole program refinement process as a composition of abstract functions. This paper designs a recurrence-based algorithm design language (Radl+) and implements a software prototype to map Radl+ algorithms into executable Haskell programs. To prove the feasibility and efficiency of this method, this paper transforms the polynomial multiplication problem from a specification into an executable Haskell program. This case shows that compared with existing approaches, the proposed method can simplify the transformation steps and reduce the number of lines of generated code from 38 to 10.

Topological, Spectroscopic and Energetic Properties of Cycloparaphenylene Series

Carbon nanorings are macrocyclic aromatic hydrocarbons that represent the segments of carbon nanotubes and have received considerable attention due to their novel nanoelectronic and photophysical capabilities. The cycloparaphenylene (CPP), a carbon nanoring with a ring-like structure comprising of only benzene units, has been the subject matter of several studies. The carbon nanoring structures bearing polycyclic aromatic hydrocarbons such as hexabenzocoronene and pyrene molecules are also found to exhibit exciting characteristics and applications. This in turn evolves a wide range of novel chemical structures whose behavior and attributes are to be studied and explored. Molecular descriptors help us to study such novel nanostructures by providing a greater understanding of their unique properties, whereas such characterization remains challenging due to their ring-shaped arrangement. In this study, we investigate cycloparaphenylene series and show that such structures are not bounded by the partial cube family. We also derive the exact analytic expressions of degree, distance and closeness-related descriptors. We have also computed the thermodynamic and kinetic stabilities of the three structures using the graph spectra, resonance energies, HOMO-LUMO gaps and Kekulé counts obtained from the graph spectra and combinatorial methods. Whereas the CPPhbc[5] structure is thermodynamically more stable, the CPPyr[9] is kinetically more stable. Furthermore, we have employed combinatorial generating function methods to compute the NMR and ESR spectra of the title compounds.

Quasi-exactly solvable extensions of the Kepler–Coulomb potential on the sphere

We consider a family of extensions of the Kepler–Coulomb potential on a d-dimensional sphere and analyze it in a deformed supersymmetric framework, wherein the starting potential is known to exhibit a deformed shape invariance property. We show that the members of the extended family are also endowed with such a property, provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build quasi-exactly solvable potentials with known ground and first-excited states. Some explicit results are obtained for the first three members of the family. We then use a generating function method, wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W+(r) [and its accompanying function W−(r)]. From the results obtained for the latter for the first three family members, we propose some formulas for W±(r) valid for the mth family member, depending on m+1 constants a0, a1, …, am. Such constants satisfy a system of m+1 linear equations. Solving the latter allows us to extend the results up to the seventh family member and then to formulate a conjecture giving the general structure of the ai constants in terms of the parameters of the problem.

Computational Triangulation in Mathematics Teacher Education

The paper is written to demonstrate the applicability of the notion of triangulation typically used in social sciences research to computationally enhance the mathematics education of future K-12 teachers. The paper starts with the so-called Brain Teaser used as background for (what is called in the paper) computational triangulation in the context of four digital tools. Computational problem solving and problem formulating are presented as two sides of the same coin. By revealing the hidden mathematics of Fibonacci numbers included in the Brain Teaser, the paper discusses the role of computational thinking in the use of the well-ordering principle, the generating function method, digital fabrication, difference equations, and continued fractions in the development of computational algorithms. These algorithms eventually lead to a generalized Golden Ratio in the form of a string of numbers independently generated by digital tools used in the paper.

Two Novel Classes of Arbitrary High-Order Structure-Preserving Algorithms for Canonical Hamiltonian Systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.