The main purpose of this paper is to introduce a new class of $q$-Hermite-Fubini numbers and polynomials by combining the $q$-Hermite polynomials and $q$-Fubini polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers. Also, we establish some relationships for $q$-Hermite-Fubini polynomials associated with $q$-Bernoulli polynomials, $q$-Euler polynomials and $q$-Genocchi polynomials and $q$-Stirling numbers of the second kind.

In this paper, we introduce a new class of $(p,q)$-analogue type of Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we consider some relationships for $(p,q)$-Fubini polynomials associated with $(p,q)$-Bernoulli polynomials, $(p,q)$-Euler polynomials and $(p,q)$-Genocchi polynomials and $(p,q)$-Stirling numbers of the second kind.

In the present paper, we introduce (p, q)-extension of Apostol type Frobenius-Euler polynomials and numbers and investigate some basic identities and properties for these polynomials and numbers, including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, explicit formulas and relations for these polynomials and numbers. Moreover, we discover (p, q)-extensions of Carlitz's result [L. Carlitz, Mat. Mag., 32 (1959), 247–260] and Srivastava and Pintér addition theorems in [H. M. Srivastava, A. Pinter, Appl. Math. Lett., 17 (2004), 375–380].

In this paper, we introduce the partially degenerate Changhee-Genocchi polynomials and numbers and investigated some identities of these polynomials. Furthermore, we investigate some explicit identities and properties of the partially degenerate Changhee-Genocchi arising from the nonlinear dierential equations.

Recently, Kim-Kim [13] have introduced polyexponential functions as an inverse to the polylogarithm functions, and constructed type 2 poly-Bernoulli polynomials. They have also introduced unipoly functions attached to each suitable arithmetic function as a universal concept. Inspired by their work, in this paper, we introduce a new class of the Frobenius-Genocchi polynomials. We derive the diverse formulas and identities covering some summation formulas, derivative formula and correlations with Bernoulli polynomials and numbers, Stirling numbers of the both kinds, degenerate Frobenius-Genocchi polynomials and degenerate Frobenius-Euler polynomials. Moreover, by using the unipoly function as following Kim-Kim's work in <cite>Kim1</cite>, we consider degenerate unipoly-Frobenius-Genocchi polynomials and investigate some formulas and relationships with Daehee numbers, degenerate Frobenius-Genocchi numbers and Stirling numbers of the first kind. Finally, we obtain an Gaussian integral representation of the Frobenius-Genocchi polynomials in terms of the 2-variable Hermite polynomials.

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

Motivated by Kurt's work [Filomat 30 (4) 921-927, 2016], we first consider a class of a new generating function for (p, q)-analog of Apostol type polynomials of order a including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order a. By making use of their generating function, we derive some useful identities. We also introduce (p, q)-analog of Stirling numbers of second kind of order v by which we construct a relation including aforementioned polynomials.

In this paper, we investigate a new $q$-analogue of the higher order degenerate Changhee polynomials and numbers, which are called the Witt-type formula for the $q$-analogue of degenerate Changhee polynomials of order $r$. We can derive some new interesting identities related to the degenerate $(h,q)$-Changhee polynomials and numbers.

In this paper, we investigate a new $q$-analogue of the higher order degenerate Changhee polynomials and numbers, which are called the Witt-type formula for the $q$-analogue of degenerate Changhee polynomials of order $r$. We can derive some new interesting identities related to the degenerate $(h,q)$-Changhee polynomials and numbers.

In this article, a new class of $q$-Hermite based unified Apostol type polynomials is introduced by means of generating function and series representation. Several important formulas and recurrence relations for these polynomials are derived via different generating methods. We also introduce $q$-analog of Stirling numbers of second kind of order $\nu$ by which we construct a relation including aforementioned polynomials.

In [18], Kim et al. introduced the degenerate poly-Bernoulli polynomials by using polyexponential function. In this paper, we study the degenerate poly-Frobenius-Genocchi polynomials, which are called the type 2 degenerate poly-Frobenius-Genocchi polynomials, by means of polyexponential function. Then, we derive some useful relations and properties. We derive type 2 degenerate poly-Frobenius-Genocchi polynomials equal a linear combination of the degenerate Frobenius-Genocchi polynomials and Stirling numbers of the first kind. Furthermore, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.

In this article, authors introduce a new class of degenerate Hermite poly-Genocchi polynomials and give some identities of these polynomials related to the Stirling numbers of the second kind. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of degenerate Hermite poly-Bernoulli numbers and polynomials studied by Khan [8].

In this paper, we consider the zero distributions of q-shift monomi-als and difference polynomials of meromorphic functions with zero order, that extends the classical Hayman results on the zeros of differential poly-nomials to q-shift difference polynomials. We also investigate problem of q-shift difference polynomials that share a common value.

This article aims to identify the generating function of modi…ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi…ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi…ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

In this paper, we study the p-adic integral representation on Z_p of q-Bernoulli numbers arising from two variable q-Bernstein polynomials and investigate some properties for the q-Bernoulli numbers. In addition, we give some new identities of q-Bernoulli numbers.

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

The main goal of the present paper is to construct some families of the Carlitz’s q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on ℤ_p. We then define the modified degenerate Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz’s q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz’s q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.

In this paper, we introduce a new generalization of the Hermite polynomials via (p, q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de…ne a (p, q)-analogue of the Bernstein polynomials and acquire their some formulas. We then provide some (p, q)-hyperbolic representations of the (p, q)-Bernstein polynomials. In addition, we obtain a correlation between (p, q)-Hermite polynomials and (p, q)-Bernstein polynomials.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Z_p of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on ℤ_{p} of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.

In this paper, we derive multifarious relationships among the two types of higher order q-Daehee polynomials and p-adic gamma function via Mahler theorem. Also, we compute some weighted p-adic q-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and the q-Bernoulli polynomials of order k.

Titania is one of the most comprehensively studied nanostructures due to its widespread applications in production of catalytic, gas- sensing and corrosion- resistance materials [1]. M-polynomial of nanotubes has been vastly investigated as it produces many degree-based topological indices which are numerical parameters capturing structural and chemical properties. These indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity and other properties of molecules are correlated with their structure like boiling point, stability, strain energy etc of chemical compounds. In this paper, we determine M-polynomials of single-walled titanium (SW TiO2) nanotubes and recover important topological degree based indices of them to theoretically judge these nanotubes. We also use Maple to plot surfaces associated to different types of single-walled titanium (SW TiO2) nanotubes.

M-polynomial of different molecular structures helps to calculate many topological indices. A topological index of graph G is a numerical parameter related to G which characterizes its molecular topology and is usually graph invariant. In the field of quantitative structure-activity (QSAR) quantitative structure-activity structure-property (QSPR) research, theoretical properties of the chemical compounds and their molecular topological indices such as the Zagreb indices, Randic index, Symmetric division index, Harmonic index, Inverse sum index, Augmented Zagreb index, multiple Zagreb indices etc. are correlated. In this report, we compute closed forms of M-polynomial, first Zagreb polynomial and second Zagreb polynomial of Octagonal network. From the M-polynomial we recover some degree-based topological indices for Octagonal network. Moreover, we give a graphical representation of our results.

In the present paper, the (p,q)-Hermite based Apostol type Frobenius-Euler polynomials and numbers are firstly considered and then diverse basic identities and properties for the mentioned polynomials and numbers, including addition theorems, difference equations, integral representations, derivative properties, recurrence relations. Moreover, we provide summation formulas and relations associated with the Stirling numbers of the second kind.

This paper aims to unify q-derivative/q-integral and h-derivative/h-integral in a single definition, this will be called q−h-derivative/q−h-integral. These notions are further extended on finite interval [a,b] in the form of left and right q−h-derivatives and q−h-integrals. Some inequalities are studied for q−h-integrals which are directly connected with well known results. In diverse fields of science and engineering the theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified by using the concept of q−h-derivative/q−h-integral.

In this paper, we investigate several relations for p-adic gamma function by means of their Mahler expansion and fermionic p-adic q-integral on ℤ_{p}. We also derive two fermionic p-adic q-integrals of p-adic gamma function in terms of q-Boole polynomials and numbers. Moreover, we discover fermionic p-adic q-integral of the derivative of p-adic gamma function. We acquire a representation for the p-adic Euler constant by means of the q-Boole polynomials. We finally develop a novel, explicit and interesting representation for the p-adic Euler constant including Stirling numbers of the first kind.

An active power dispatch method for a microgrid (MG) with multi-type loads, renewable energy sources (RESs) and distributed energy storage devices (DESDs) is the focus of this paper. The MG operates in a grid-connected model, and distributed power sources contribute to the service for load demands. The outputs of multiple DESDs are controlled to optimize the active power dispatch. Our goal with optimization is to reduce the economic cost under time-of-use (TOU) price, and to adjust the excessively high or low load rate of distributed transformers (DTs) caused by the peak-valley demand and load uncertainties. To simulate a practical environment, the stochastic characteristics of multi-type loads are formulated. The transition matrix of system state is provided. Then, a finite-horizon Markov decision process (FHMDP) model is established to describe the dispatch optimization problem. A learning-based technique is adopted to search the optimal joint control policy of multiple DESDs. Finally, simulation experiments are performed to validate the effectiveness of the proposed method, and the fuzzification analysis of the method is presented.

Many corporations have now revised their value statements to showcase their commitment to sustainable development and green business due to the pressing issues of climate and environmental damage. Despite the importance of value statements that can direct companies to achieve sustainable goals, what kind of values can be perceived as organizational orientations to sustainability remains unknown. This study explores how the general public views the inclusion of sustainable components in value statements from the listed companies in Hong Kong by using Q-sort methodology (n = 30). Analyzing the final sample of 40 value statements extracted from 27 listed companies, we identified three factors related to sustainability, namely, Empowering business process management, Empowering customer-centric excellence, and Empowering sustainable progress. This study provides an additional component to the traditional triple bottom line by suggesting that internal operation of a company is an emerging idea to understand corporate sustainability.

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here the coefficients involve terminating hypergeometric functions ₂F₁ and these representations are obtained by explicit computations.

This paper is concerned with representing sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials which involve the hypergeometric functions ${}_1 F_1$ and ${}_2 F_1$.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.

Here we consider sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials which involve the hypergeometric function ₃F₂.

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

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.

In the paper, by virtue of the Faá di Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds.

The prim objective of commenced article is to determine q-sumudu transforms of a product of unified family of q-polynomials with basic (or q-) analogue of fox’s H-function and q-analog of I-functions. Specialized cases of the leading outcome are further evaluated as q-sumudu transform of general class of q-polynomials and q-sumudu transforms of the basic analogues of Fox’s H-function and I-functions.

Level index was introduced in 2017 for rooted trees which is a component of Gini index. In the origin, Gini index is a tool for economical investigations but Balaji and Mahmoud defined the graph theoretical applications of this index for statistical analysis of graphs. Level index is an important component of Gini index. In this paper we define a new graph polynomial which is called level polynomial and calculate the level polynomial of some classes of trees. We obtain some interesting relations between the level polynomials and some integer sequences.

In this paper, we firstly consider extended degenerate central factorial numbers of the second kind and provide some properties of them. We then introduce unified degenerate central Bell polynomials and numbers and investigate many relations and formulas including summation formula, explicit formula and derivative property. Moreover, we derive several correlations for the fully degenerate central Bell polynomials associated with the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers.

In this paper, we represent the generating function of w-torsion Fubini polynomials by fermionic p-adic integral on Z_p. Then we consider a quotient of such fermionic p-adic integrals on Z_p, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials.

We study on the sums of powers of consequtive integers and alternating sums of power of consequtive integers. We derive many identities and correlations including Bernoulli, Euler and Genocchi polynomials and numbers.

This paper includes some new investigations and results for post quantum calculus, denoted by (p,q)-calculus. A chain rule for (p,q)-derivative is developed. Also, a new (p,q)-analogue of the exponential function is introduced and some its properties including the addition property for (p,q)-exponential functions are investigated. Several useful results involving (p,q)-binomial coefficients and (p,q)-antiderivative are discovered. At the final part of this paper, (p,q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.

We construct a novel family of summation-integral type hybrid operators in terms of shape parameter $\alpha\in \lbrack 0,1]$ in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre's K-functional, Lipschitz class, and second-order modulus of smoothness.

We propose a new statistical analysis of the Acoustic Emissions (AE) produced in a series of triaxial deformation experiments leading to fractures and failure of two different rocks, namely, Darley Dale Sandstone (DDS) and AG Granite (AG). By means of q-statistical formalism, we are able to characterize the pre-failure processes in both types of rocks. In particular, we study AE inter-event time and AE inter-event distance distributions. Both of them can be reproduced with q-exponential curves, showing universal features which are observed here for the first time and could be important in order to understand more in detail rock fractures dynamics.

In this article, the Legendre-Gould Hopper polynomials are combined with Sheffer sequences to introduce certain mixed type special polynomials. Generating functions, differential equations and certain other properties of Legendre-Gould Hopper based Sheffer polynomials are derived. Further, operational and integral representations providing connections between these polynomials and known special polynomials are established. Certain identities and results for some members of these new mixed polynomials are also obtained. Finally, the determinantal definitions of Legendre-Gould Hopper based Sheffer polynomials are also given.

In this paper, we introduce a new class of degenerate Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of degenerate Hermite-Fubini numbers and polynomials by using generating functions.

In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

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 Faà 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.

We use a recently proposed formulation of quantum mechanics based, not on potential functions but rather, on orthogonal energy polynomials. In this context, the most important building block of a quantum mechanical system, which is the wavefunction at a given energy, is expressed as pointwise convergent series of square integrable functions in configuration space. The expansion coefficients of the series are orthogonal polynomials in the energy; they contain all physical information about the system. No reference is made at all to the usual potential function. We consider, in this new formulation, few representative problems at the level of undergraduate students who took at least two courses in quantum mechanics and are familiar with the basics of orthogonal polynomials. The objective is to demonstrate the viability of this formulation of quantum mechanics and its power in generating rich energy spectra illustrating the physical significance of these energy polynomials in the description of a quantum system. To assist students, partial solutions are given in an appendix as tables and figures.

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

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 Faà 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.

The prevalence of differential equations as a mathematical technique has refined the fields of control theory and constrained optimization due to the newfound ability to accurately model chaotic, unbalanced systems. However, in recent research, systems are increasingly more nonlinear and difficult to model using Differential Equations only. Thus, a newer technique is to use policy iteration and Reinforcement Learning, techniques that center around an action and reward sequence for a controller. Reinforcement Learning (RL) can be applied to control theory problems since a system can robustly apply RL in a dynamic environment such as the cartpole system (an inverted pendulum). This solution successfully avoids use of PID or other dynamics optimization systems, in favor of a more robust, reward-based control mechanism. This paper applies RL and Q-Learning to the classic cartpole problem, while also discussing the mathematical background and differential equations which are used to model the aforementioned system.

Motivated by Kim-Kim [19] introduced the new type of degenerate poly- Bernoulli polynomials by means of the degenerate polylogarithm function. In this paper, we define the degenerate poly-Frobenius-Euler polynomials, called the new type of degenerate poly-Frobenius-Euler polynomials, by means of the degenerate polylogarithm function. Then, we derive explicit expressions and some identities of those numbers and polynomials.

In this paper, we introduce a new class of Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions.

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.

Discrete Double Hilbert Exponential sums along polynomials. This is preprint version (not peer reviewed) And we now check the proof of (ii) of Lemma 3.2 whether it is really true.

A unified presentation of a class of Humbert’s polynomials in two variables which generalizes the well known class of Gegenbauer, Humbert, Legendre, Chebycheff, Pincherle, Horadam, Kinnsy, Horadam-Pethe, Djordjević , Gould, Milovanović and Djordjević, Pathan and Khan polynomials and many not so called ’named’ polynomials has inspired the present paper and the authors define here generalized Humbert-Hermite polynomials of two variables. Several expansions of Humbert- Hermite polynomials, Hermite-Gegenbaurer (or ultraspherical) polynomials and Hermite- Chebyshev polynomials are proved.

The main aim of this paper is to investigate multifarious properties and relations for the gamma distribution. The approach to reach this purpose will be introducing a special polynomial including gamma distribution. Several formulas covering addition formula, derivative property, integral representation and explicit formula are derived by means of the series manipulation method. Furthermore, two correlations including Bernoulli and Euler polynomials for gamma distribution polynomials are provided by utilizing of their generating functions.

Given the AdS/CFT relationship, the study of higher-dimensional AdS black holes is extremely important. Furthermore, since the restriction derived from $f(Q)$'s field equations which prevents it from deriving spherically symmetric black hole solutions, the result is either $Q'=0$ or $f_{QQ}=0$. We derive static anti-de-Sitter black holes in $D$-dimensions using the cylindrical coordinate in the framework of the cubic form of $f(Q)$ gravity, where the coincident gauge condition is imposed \cite{Heisenberg:2023lru,Maurya:2023muz}. Within this study we are going to use the power-law ansatz which is the most viable according to observations where $f(Q)=Q+\frac{1}2\gamma Q^2+\frac{1}3\gamma Q^3-2\Lambda$, and the condition $D \geq 4$ is satisfied. Since these solutions lack a general relativity limit, they belong to a brand-new solution class, the characteristics of which are derived only from the non-metricity $Q$ modification. We investigate the singularities of the solutions by computing the values of different curvature and non-metricity invariants. We discover that the solutions indeed have a central singularity, but because of the f(Q) impact, it is softer than in the typical general relativity case. In conclusion, we compute thermodynamic parameters such as Gibbs free energy, Hawking temperature, and entropy. These thermodynamic calculations confirm that our model is stable.

In this paper we introduce the hypo-q-norms on a Cartesian product of inner product spaces. A representation of these norms in terms of inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar Shisha-Mond, Birnacki et al., Grüss type inequalities, Boas-Bellman and Bombieri type inequalities are also given.

Attitude control of a novel regional truss-braced wing aircraft with low stability characteristics is addressed in this paper using Reinforcement Learning (RL). In recent years, RL has been increasingly employed in challenging applications, particularly, autonomous flight control. However, a significant predicament confronting discrete RL algorithms is the dimension limitation of the state-action table and difficulties in defining the elements of the RL environment. To address these issues, in this paper, a detailed mathematical model of the mentioned aircraft is first developed to shape an RL environment. Subsequently, Q-learning, the most prevalent discrete RL algorithm will be implemented in both the Markov Decision Process (MDP), and Partially Observable Markov Decision Process (POMDP) frameworks to control the longitudinal mode of the air vehicle. In order to eliminate residual fluctuations that are a consequence of discrete action selection, and simultaneously track variable pitch angles, a Fuzzy Action Assignment (FAA) method is proposed to generate continuous control commands using the trained Q-table. Accordingly, it will be proved that by defining an accurate reward function, along with observing all crucial states (which is equivalent to satisfying the Markov Property), the performance of the introduced control system surpasses a well-tuned Proportional–Integral–Derivative (PID) controller.

We introduce the lower chaos grade of a real-valued function F defined on the Markov triple (E,μ,Γ), where μ is a probability measure and Γ is the carré du champ operator. As an application of this concept, we obtain the better estimate of the four moments theorem for Markov diffusion generators worked by Bourguin et al (2019). For our purpose, we need to find the largest number except zero in the set of eigenvalues corresponding to its eigenfunction in the case where the square of a random variable F, coming from a Markov triple structure, can be expressed as a sum of eigenfunctions, We give some examples of eigenfunctions of the diffusion generators such as Ornstein-Uhlenbeck, Jacobi and Romanovski-Routh. In particular, two bounds, called the four moments theorem and fourth moment theorem respectively, will be provided for the normal approximation of the case where a random variable F comes from eigenfunctions of a Jacobi generator.

Let $S=\left(s_1, \ldots, s_n\right)$ be a finite sequence of integers. Then $S$ is a Gilbreath sequence of length $n$, $S\in\mathbb{G}_n$, iff $s_1$ is even or odd and $s_2, \ldots, s_n$ are respectively odd or even and $\min\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\leq s_{m+1}\leq\max\mathbb{K}_{\left(s_1, \ldots, s_m\right)}\forall m\in\left[\left.1, n\right)\right.$. This, applied to the order sequence of prime number $P$, defines Gilbreath polynomials and two integer sequences A347924 \cite{oeisA347924} and A347925 \cite{oeisA347925} which are used to prove that Gilbreath conjecture $GC$ is implied by $p_n-2^{n-1}\leqslant\mathcal{P}_{n-1}\left(1\right)$ where $\mathcal{P}_{n-1}\left(1\right)$ is the $n-1$-th Gilbreath polynomial at 1.

In this paper, we introduce a new type of degenerate poly-Fubini polynomials and numbers, are called degenerate poly-Fubini polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we also consider the degenerate unipoly-Fubini polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

In this paper, we construct the degenerate poly-Fubini polynomials, called the type 2 degenerate poly-Fubini polynomials, by using the modified degenerate polyexponential function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we introduce type 2 degenerate unipoly- Fubini polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities for those polynomials. Furthermore, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

In this paper, we introduce a new class of Laguerre-based Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Laguerre-based Hermite-Fubini numbers and polynomials by using generating functions.

The topological index is a numerical quantity based on the characteristics of various invariants or molecular graph. For ease of discussion, these indices are classified according to their logical derivation from topological invariants rather than their temporal development. Degree based topological indices depends upon the degree of vertices. This paper computes degree based topological indices of Bismuth Tri-Iodide chains and sheets with the help of M-polynomial.

In the study of QSAR/QSPR, due to high degree of predictability of pharmaceutical properties, the eccentric-connectivity index has very important place among the other topological descriptors, In this paper, we compute the exact formulas of eccentric-connectivity index and its corresponding polynomial, total eccentric-connectivity index and its corresponding polynomial, first Zagreb eccentricity index, augmented eccentric-connectivity index, modified eccentric-connectivity index and its corresponding polynomial for a class of phosphorus containing dendrimers.

Here, we describe the bicomplex (p,q)- Fibonacci numbers and the bicomplex (p,q)- Fibonacci quaternions that are based on these numbers and give some of their equations, including the Binet formula, generating function, Catalan, Cassini, d’Ocagne’s identities, and some summation formulas for both of them. Finally, we create a matrix for bicomplex (p,q)- Fibonacci quaternions, and we obtain a determinant of a special matrix that gives the terms of that quaternion.

We propose a correspondence between partition functions of ideal gases consisting of both bosons and fermions and algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sides sequences ℓ1(Z0). Such an approach allows us to interpret some combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence on ℓ1(Z0) induced by the supersymmetric polynomials, and semiring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semiring structures on this set and discuss their possible physical interpretations.

In this paper we consider Changhee polynomials of type two, which are motivated from the recent work of Kim and Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmtery for the fermionic p-adic integral on Zp.

We investigate the value distribution of dierence polynomials of entire and meromorphic functions, which can be viewed as the Hayman's conjecture. And we also study the uniqueness of dierence polynomials sharing a common value.

Reinforcement learning (RL) is an extensively applied control method for the purpose of designing intelligent control systems to achieve high accuracy as well as better performance. In the present article, the PID controller is considered as the main control strategy for brushless DC (BLDC) motor speed control. For better performance, the fuzzy Q-learning (FQL) method as a reinforcement learning approach is proposed to adjust the PID coefficients. A comparison with the adaptive PID (APID) controller is also performed for the superiority of the proposed method, and the findings demonstrate the reduction of the error of the proposed method and elimination of the overshoot for controlling the motor speed. MATLAB/SIMULINK has been used for modeling, simulation, and control design of the BLDC motor.

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

Dermal pigmentation is a challenging condition due to relative resistance to medical treatment and requirement of multiple sessions with modalities like chemical peels and lasers. There is inadequate literature regarding the efficacy and safety of lasers for dermal pigmentation in Indian skin. One hundred and twenty two patients were enrolled and divided into 4 groups: a) Dermal melasma b) Acquired dermal macular hyperpigmentation c) Nevus and d) Tattoo. Sequential combination of various lasers were used with different protocols for 4 different groups like ablative pixel Er:YAG, intense pulse light based technologies, Q-Switched and Alma Hybrid lasers for treatment of dermal pigmentation. A total of 4-10 sessions were conducted at three weekly intervals. Results were evaluated by comparing photographs at baseline, one month after the last session and then at 6 months interval to assess the relapse. All groups showed a statistically significant reduction in pigmentation. The mean treatment duration was 5-7 months for tattoo, lichen planus pigmentosus and melasma with 6-8 number of sessions. The mean treatment duration for nevus was 7-9 months with 8-10 sessions. Lasers are safe and effective modalities in Indian skin type for treatment of dermal pigmentation. Treatment period and relapse of dermal pigmentation may be reduced by combining resurfacing and chromophore selective lasers along with adequate dietary and lifestyle modifications to minimize complications and recurrence rates.

Motivation by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim [9], in the present paper, we consider a new class of new generating function for the Fubini polynomials, called the type 2 poly-Fubini polynomials by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Fubini polynomials equal a linear combination of the classical of the Fubini polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Fubini polynomials and Bernoulli polynomials of order r. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim [9]. We introduce the type 2 unipoly-Fubini polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Fubini polynomials and the classical Fubini polynomials.

In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers.

Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.

In this paper, we investigate the new degenerate Daehee polynomials and numbers which are called the degenerate Daehee polynomials of the second kind, and derive some new and interesting identities and properties of those polynomials.

Rickettsia and Coxiella burnetii are zoonotic tick-borne pathogens that can cause febrile illnesses with or without other symptoms in humans but may cause subclinical infections in animals. There are only a few reports on the occurrence of these pathogens in cattle and water buffaloes in Southeast Asia, including the Philippines. In this study, molecular detection of Rickettsia spp. and C. burnetii in the blood and Rhipicephalus (Boophilus) microplus ticks of cattle and water buffaloes from five provinces in Luzon Island of the Philippines was done. A total of 620 blood samples of cattle and water buffaloes and 206 tick samples were collected and subjected to DNA extraction. After successful amplification of control genes, nested PCR was performed to detect gltA of Rickettsia and com1 of C. burnetii. No samples were positive for Rickettsia while 10 (cattle – 7, water buffaloes - 3) or 1.6% of blood and 5 or 1.8% of tick samples were C. burnetii-positive. Sequence analysis of the positive amplicons showed 99-100% similarity to reported C. burnetii isolates. This molecular evidence on the occurrence of C. burnetii in Philippine ruminants and cattle ticks and its zoonotic nature should prompt further investigation and surveillance to facilitate its effective control.

In this paper, we revisit the q-states clock model for small systems. We present results for the thermodynamics of the q-states clock model from $q=2$ to $q=20$ for small square lattices $L \times L$, with L ranging from $L=3$ to $L=64$ with free-boundary conditions. Energy, specific heat, entropy and magnetization are measured. We found that the Berezinskii-Kosterlitz-Thouless (BKT)-like transition appears for $q>5$ regardless of lattice size, while the transition at $q=5$ is lost for $L<10$; for $q\leq 4$ the BKT transition is never present. We report the phase diagram in terms of $q$ showing the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at a critical temperature T$_1$ for small systems which turns into a transition from the FM to the BKT phase for larger systems, while a second phase transition between the BKT and the PM phases occurs at T$_2$. We also show that the magnetic phases are well characterized by the two dimensional (2D) distribution of the magnetization values. We make use of this opportunity to do an information theory analysis of the time series obtained from the Monte Carlo simulations. In particular, we calculate the phenomenological mutability and diversity functions. Diversity characterizes the phase transitions, but the phases are less detectable as $q$ increases. Free boundary conditions are used to better mimic the reality of small systems (far from any thermodynamic limit). The role of size is discussed.

Undoubtedly the Wageningen B-screw Series is the most widely used systematic propeller series. It is very popular to preselect propeller dimensions during the preliminary design stage before performing a more thorough optimization, but in the smaller end of the market it is often used to merely select the final propeller. Over time the originally measured data sets were faired and scaled to a uniform Reynolds number of 2·10⁶ to increase the reliability of the series. With the advent of the computer, polynomials for the thrust and torque values were calculated based on the available data sets. The measured data are typically presented in the well-known open-water curves of thrust and torque coefficients K_T and K_Q versus the advance coefficient J. Changing the presentation from these diagrams to efficiency maps reveals some unsuspected and surprising behaviours, such as multiple optima when optimizing for efficiency or even no optimum at all for certain conditions. These artefacts get more pronounced at higher pitch to diameter ratios and low blade numbers. The present work builds upon the paper presented by the author at the AMT’17 and smp’19 conferences and now includes the extended efficiency maps, as suggested by Danckwardt, for all propellers of the Wageningen B-screw Series.

We introduce the Hermite based poly-Bernoulli polynomials with a q parameter and give someof their basic properties including not only addition property, but also derivative properties and integralrepresentations. We also de.ne the Hermite based λ -Stirling polynomials of the second kind, and thenprovide some relations. Moreover, we derive several correlations and identities including the Hermite-Kampéde Fériet (or Gould-Hopper) family of polynomials, the Hermite based poly-Bernoulli polynomials with a qparameter and the Hermite based λ -Stirling polynomials of the second kind.

This paper presents the symmetries of differential equations associated with one-variable and Bivariate Hermite polynomials by proposing a representation of Lie algebra for these differential operators. Applying the Baker-Campbell-Hausdorff formula to these algebras, results in new relations and generating functions in one-variable and Bivariate Hermite polynomials. A general form of representation for other orthogonal polynomials such as Laguerre polynomials is introduced.

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

Graph theory plays a crucial role in modeling and designing of chemical structure or chemical network. Chemical Graph theory helps to understand the molecular structure of molecular graph. The molecular graph consists of atoms as vertices and bonds as edges. Topological indices capture symmetry of molecular structures and give it a mathematical language to predict properties such as boiling points, viscosity, the radius of gyrations etc. In this article, we study the chemical graph of carbon Crystal structure of graphite and cubic carbon and compute several degree-based topological indices. Firstly we compute M-Polynomials of these structures and then from these M-polynomials we recover nine degree-based topological indices.

Coenzyme Q (CoQ) is a lipidic compound widely distributed in nature with crucial functions in metabolism, protection against oxidative damage and ferroptosis, and other processes. CoQ biosynthesis is a conserved and complex pathway involving several proteins. COQ2 is a member of the UbiA family of transmembrane prenyltransferases that catalyzes the condensation of the head and tail precursors of CoQ, a key step in the process because its product is the first intermediate that will be modified in the head by the next component of the synthesis process. Mutations in this protein have been linked to primary CoQ deficiency in humans, a rare disease predominantly affecting organs with a high energy demand. The reaction catalyzed by COQ2 and its mechanism are still unknown. Here we aimed at clarifying COQ2 reaction by exploring possible substrate binding sites using a strategy based on homology, comprising the identification of ligand-bound homologs with solved structures available in the Protein Data Bank (PDB) and their subsequent structural superposition to the AlphaFold predicted model for COQ2. The results highlight some residues located on the central cavity or the matrix loops that may be involved in substrate interaction, some of them mutated in primary CoQ deficiency patients. Furthermore, we analyze the structural modifications introduced by the pathogenic mutations found in humans. These findings shed new light on the understanding of COQ2 function and, thus, CoQ biosynthesis and pathogenicity of primary CoQ deficiency.

New q-series in the spirit of Jacobi have been found in a publication rst published in 1884 written in Russian and translated into English in 1928. This work was found by chance and appears to be almost totally unknown. From these entirely new q-series, fresh lattice sums have been discovered and are presented here.

For evaluating probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based upon the Gauss-Laplace law. The latter will be considered here as an element of the newly introduced family of (p,q)-spherical distributions. Based upon a suitably defined non-Euclidean arc-length measure on (p,q)-circles we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly like with elliptically contoured distributions and more general homogeneous star-shaped ones. This is demonstrated at hand of a generalization of the Box-Muller simulation method. En passant, we prove an extension of the sector and circle number functions.

In this paper, we construct the degenerate poly-Frobenius-Genocchi polynomials, called the type 2 degenerate poly-Frobenius-Euler polynomials, by means of polyexponential function. We derive explicit expressions and some identities of those polynomials. In the last section, we introduce type 2 degenerate unipoly-Frobenius-Genocchi polynomials by means of unipoly function and derive explicit multifarious properties.

We introduce a new kind of extended Hermite-based Frobenius type Eulerian polynomials and then derive diverse explicit and implicit summation equations including some symmetric formulas by utilizing series manupulation method. Multifarious summation formulas and identities given earlier for some well known polynomials such as Eulerian polynomials and Frobenius type Eulerian polynomials are generalized.

The main aim of this paper is to set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on ℤ_{p}. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.

By utilizing a generalized version of the Madelung quantum-hydrodynamic framework that incorporates noise, we derive a solution using the path integral method to investigate how a quantum superposition of states evolves over time. This exploration seeks to comprehend the process through which a stable quantum state ultimately emerges in the presence of fluctuations induced by the noisy gravitational background. The model defines the conditions that give rise to a limited range of interaction for the quantum potential, allowing for the existence of coarse-grained classical descriptions at a macroscopic level. The theory uncovers the smallest attainable level of uncertainty in an open quantum system and examines its consistency with the localized behavior observed in large-scale classical systems. The research delves into connections and similarities with decoherence theory and the Copenhagen interpretation of quantum mechanics. Additionally, it assesses the potential consequences of wave function decay on the measurement of photon entanglement. To validate the proposed theory, an experiment involving entangled photons transmitted between detectors on the Moon and Mars is discussed. Finally, the findings of the theory are applied to the creation of larger Q-bits systems at elevated temperatures.

Background: This literature review investigates qualitative and quantitative uses of indocyanine green (ICG) in surgical treatment, as this novel technique has become increasingly popular in ORs (operating rooms) worldwide due to its many advantages. Method: An extensive literature review was performed by searching for relevant terms in 5 international databases (www.pubmed.gov, www.sciencedirect.com, www.scopus.com, www.Oxfordjournals.com, www.reaxys.com). The results of this search were summarized into the main advantages of employing ICG in colorectal surgery as follows: a) intraoperative fluorescence angiography capability; b) fluorescence-guided identification of lymph node involvement in colorectal cancer and the sentinel lymph node method; c) positive fluorescence flagging of a liver tumour as small as "only" 200 tumour cells; d) facilitation of fistula diagnosis; and e) tumour tattooing. Moreover, in addition to qualitative intraoperative ICG use, this technique can be combined with quantitative methods and parameters, such as maximum intensity, relative maximum intensity, and different inflow parameters (time-to-peak, slope, and t1/2max). The conclusion of this article is that fluorescence surgery with ICG aided by near-infrared (NIR) light is a relatively recent technology, an advantage of which is improving the specific anatomic and functional information provided, thus allowing more comprehensive and safer tumour removal as well as preservation of important structures.

Kim(2015) introduced the degenerate Cauchy numbers of the second kind and gave some identities of them. In this paper, by using the modified polyexponential functions which are introduced by Kim-Kim(2019), we define the degenerate poly-Cauchy polynomials of the second kind and investigate some properties of them. Furthermore, we consider the degenerate unipoly-Cauchy polynomials of the second kind and discuss some properties of them.

The present analysis deals with the study of the $f(Q, T )$ theory of gravity which was recently considered by many cosmologists. In this theory of gravity, the action is taken as an arbitrary function $f(Q, T )$ where $Q$ is non-metricity and $T$ is the trace of energy-momentum tensor for matter fluid. In this study, we have taken three different forms of the function $f(Q, T )$ as $f(Q,T)=a_1Q+a_2 T$ and $f(Q,T)=a_3 Q^2+a_4 T$ and discussed some physical properties of the same. Also, we have obtained the various cosmological parameters towards Friedmann-Lemaitre-Robertson-walker (FLRW) Universe by defining the transit form of scale factor which yields the Hubble parameter in redshift form as $H(z)=\frac{H_{0}}{(\lambda+1)} \left(\lambda+ (1+z)^{\delta}\right)$. %We have obtained the approx best-fit values of model parameters using the least square method for observational constraints on available datasets like Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc., by applying the Root Mean Squared Error formula (RMSE). By applying the Root Mean Squared Error formula (RMSE), we were able to determine the approximate best-fit values of model parameters using the least square approach for observational constraints on the datasets Hubble dataset $H(z)$, Supernova dataset SNe-Ia, etc. We have observed that the deceleration parameter $q(z)$ exhibits a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from the decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$ for the approximate best-fit values of the model parameters. %For obtained approx best-fit values of model parameters we have observed that the deceleration parameter $q(z)$ shows a signature-flipping (transition) point within the range $0.623 \le z_{0} \le 1.668$ through which it changes its phase from decelerated to the accelerated expanding universe with $\omega = -1$ at $z=-1$.

In this paper, the stabilization and high efficiency of an unstable Second Harmonic Generation (SHG) of a Nd : YVO4 laser with a KTP intracavity is carried out by using a passive Q-switching crystal and a parametric modulation method (harmonic modulation) is shown. The harmonic modulation was defined as um(t) = Amsin(2π fmt), with modulation amplitude as Am(Vpp), and modulation frequency as fm(Hz), applied to the pumping of the Nd : YVO4 -KTP laser to control the amplitude and frequency of the emission, thus stabilizing the dynamic states of this laser. The promising application of this green light source is materialized when such light is necessary for high-density optics, such as in the treatment materials industry or in some aesthetic applications.

Suppose G is a graph with vertex-orbits O1,1 O2, . . . ,Ot, and jOi j denotes the cardinallity of Oi. Then OG (x) = åt 2 i=1 xjOi j is called as orbit polynomial. It is well-known that this polynomial 3 has a unique positive zero d in the interval [0, 1]. The aim of this paper is to study the specific 4 properties of this polynomial and then we determine the location of this root for several classes of 5 complex networks to compare with other graphical measures. Besides, we compare the unique 6 positive zero measure with several well-known centrality graph measures.

In this paper, we introduce a general family of Lagrange-based Apostol-type Hermite polynomials thereby unifying the Lagrange-based Apostol Hermite-Bernoulli and the Lagrange-based Apostol Hermite-Genocchi polynomials. We also define Lagrange-based Apostol Hermite-Euler polynomials via the generating function. In terms of these generalizations, we find new and useful relations between the unified family and the Apostol Hermite-Euler polynomials. We also derive their explicit representations and list some basic properties of each of them. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions.