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A class of variable‐order fractional optimal control problems (VO‐FOCPs) is solved by applying Müntz‐Legendre wavelets. Different from classical wavelets (such as Legendre and Chebyshev), the Müntz‐Legendre wavelets have an extra parameter representing the fractional order; therefore, they provide more reliable results for certain fractional calculus problems. Using the regularized beta function, the Riemann‐Liouville fractional integral operator (RLFIO) of these wavelets is precisely determined. We then transform the given VO‐FOCPs into parameter optimization problems that can be easily solved. The convergence analysis and error estimation of the proposed method are provided. Four examples are solved to illustrate the high accuracy of the approach. The method is also applied to a cancer‐obesity interaction model to analyze the interactions between the tumor, immune, normal, and fat cells when the chemotherapeutic drugs are injected into a body. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new fractional operators, MRL fractional integral, MRL fractional derivative and MC fractional derivative operators, which including generalized M-series in their kernels and give some of their fundamental properties. Then we apply Laplace, Mellin and beta integral transformations to the new fractional operators and obtain conclusions involving classical gamma and beta and modified gamma and beta functions. As examples, we also obtain similar conclusions by applying new fractional operators to the functions zλ and (1 - az)-λ. Furthermore, we present the relationships of the new fractional operators with other fractional operators in the literature. Finally, we compare the behavior of the function z² in classical Riemann-Liouville and Caputo fractional operators and MRL and MC fractional operators for orders ε = 0:2;0:4;0:6;0:8. [ABSTRACT FROM AUTHOR]

Infinite series involving Riemann's zeta and Dirichlet's lambda tails, and weighted by three harmonic-like elementary symmetric functions are examined. By means of integral representations of zeta tails together with the telescopic approach, twelve general summation theorems are established that express these series as coefficients of the bivariate beta function Beta (u , v) . By further expanding Beta (u , v) into Laurent series in u and v, several explicit summation formulae are shown as consequences. [ABSTRACT FROM AUTHOR]

This work is devoted to define new generalized gamma and beta functions involving the recently suggested seven-parameter Mittag-Leffler function, followed by a review of all related special cases. In addition, necessary investigations are affirmed for the new generalized beta function including, Mellin transform, differential formulas, integral representations, and essential summation relations. Furthermore, crucial statistical application has been realized for the new generalized beta function. [ABSTRACT FROM AUTHOR]

In this article, authors introduced the new (p, q; v)-extended Bessel-Wright function ..., some properties related to Marichev-Saigo-Maede fractional integral and derivative operators, and Caputo-type Marichev-Saigo-Maede fractional integral and derivative operators which are applied to the (p, q; v)-extended Bessel-Wright function. Some special cases such as Saigo, Riemann-Liouville and Erdeyi-Kober fractional integrals and derivative operators are obtained. In addition, applications of the new (p, q; v)-extended Bessel-Wright function ... to the fractional kinetic equations is also discussed. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to discuss some classical properties of the pRq(α, β; z) function such as integrals involving pRq(α, β; z) function and its product with some algebraic functions and higher Tanscendental function viz, Hermite polynomial, Legendre polynomial, Legendre function, Jacobi polynomial, Galue type Struve function, six summation formulas of pRq(α, β; z) function and relation between pRq(α, β; z) and pRq(α, β;-z) functions. [ABSTRACT FROM AUTHOR]

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In this paper, we present a modification of Jain-Pethe-Baskakov-Durrmeyer operators and estimate their moments. Then, we establish the uniform convergence of the proposed family of operators. Further, we use modulus of continuity and K -functional to establish local approximation behavior of these operators. Also, we compute the rate of convergence of the operators through Lipschitz class functions. In the last section, we present some quantitative results for difference of the proposed operators with Jain-Pethe operators and Durrmeyer type variant of Jain-Pethe-Baskakov operators. [ABSTRACT FROM AUTHOR]

In this article, we propose a new extension of beta function by utilizing the Bessel-Struve kernel function. Here, first we derive some basic properties of this new beta function and thereafter present a new extension of the well-known beta dispersion as an application of our proposed beta function. Additional to that we elaborate and explore one more extension of Gauss and confluent hypergeometric functions by utilizing the definition of the same. Some significant properties of the above mentioned hypergeometric functions like integral representations, differential formulae, transformation formulae, summation formulae and a generating relation are are also pointed out in a systematic manner. [ABSTRACT FROM AUTHOR]

In this study, the beta function that is encountered in computational mathematics and physics is analyzed. The correct evaluation of this function also affects the accuracy of other mathematical functions in quantum mechanical calculations. Especially in recent years, there is an interest in studies related to the beta function for zero and negative p and q integers. This study, considering the neutrix limits of the beta function, presents new relations for the numerical computation of the beta function, especially for negative integers p and q. In addition, taking into account the definition of the beta function for positive p and q integer values, an algorithm is created to calculate the function for all integer values. Finally, numerical results obtained with the help of our new recurrence relations and algorithm are presented.

In this article, authors discussed about image formulas for the some new extended hypergeometric function using elementary properties of the fractional calculus integral operators. Furthermore, using integral transforms some image formulas are also established. [ABSTRACT FROM AUTHOR]

By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard's inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as 1 (x + ∥ k − ξ ∥ α) λ + m (x , λ > 0) involving one derivative function of m-order is given. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms. the operator expressions and some particular inequalities are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new Carlson-type integral inequality with the best constant factor. The equivalent Beurling-Kjellberg type inequality and discrete form are considered. [ABSTRACT FROM AUTHOR]

By means of the weight functions, the idea of introducing parameters and the technique of real analysis related to the beta and gamma functions, a new reverse Hardy–Hilbert-type integral inequality with the homogeneous kernel as 1 (x + y) λ + m + n (λ > 0 ) involving two derivative functions of higher order is given. As applications, the equivalent statements of the best possible constant factor related to several parameters are considered, and some particular inequalities are obtained. [ABSTRACT FROM AUTHOR]

In this paper, using the modified beta function involving the generalized M-series in its kernel, we describe new extensions for the Lauricella hypergeometric functions F(r) A, F(r) B, F(r) C and F(r) D. Furthermore, we find various integral representations for the newly defined extended Lauricella hypergeometric functions. Then, we obtain solution of fractional differential equations involving new extensions of Lauricella hypergeometric functions, as examples. [ABSTRACT FROM AUTHOR]

We give the hybrid of block-pulse function and generalized Mott polynomials (HBGMP) and use it to solve delay fractional optimal control problems (DFOCPs). First, we develop a method for computing the exact formula of the Riemann–Liouville fractional integral operator of the HBGMP by using the regularized beta function. Next, the DFOCPs will be transformed into parameter optimization problems. By imposing the optimality conditions, we reduce the problem to algebraic equations. Several examples are given to show the advantages of the method. [ABSTRACT FROM AUTHOR]

Entropies are useful measures of variation. However, explicit expressions for entropies available in the literature are limited. In this paper, we provide a comprehensive collection of explicit expressions for four of the most common entropies for over sixty continuous univariate distributions. Most of the derived expressions are new. The explicit expressions involve known special functions. [ABSTRACT FROM AUTHOR]

By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality with the homogeneous kernel 1 (x + y) λ (λ > 0) involving two multiple upper-limit functions is obtained. The equivalent statements of the best possible constant factor related to the beta and gamma functions are considered. As applications, the equivalent forms and the case of a nonhomogeneous kernel are deduced. Some particular inequalities and the operator expressions are provided. [ABSTRACT FROM AUTHOR]

By means of the weight functions, the idea of introduced parameters, using the transfer formula and Hermite–Hadamard's inequality, a more accurate half-discrete multidimensional Hilbert-type inequality with the homogeneous kernel as 1 (x + | | k − ξ | | α) λ (x , λ > 0) involving one multiple upper limit function is given, which is a new application of Hilbert-type inequalities. The equivalent conditions of the best possible constant factor related to several parameters are considered. The equivalent forms the operator expressions and some particular inequalities are obtained. [ABSTRACT FROM AUTHOR]

In this short note we use the combinatorial identity of Chaundy and Bullard to prove several identities involving the Pochhammer symbol. For example, it is proven that (Y) n + 1 ∑ k = 0 m n + k k (X) k (X + Y) n + k + 1 + (X) m + 1 ∑ k = 0 n m + k k (Y) k (X + Y) m + k + 1 = 1 where the Pochhammer symbol, or the rising factorial, is denoted by (z) n. [ABSTRACT FROM AUTHOR]

In this paper, we present ̅q-Laplace transform by ̅q-integral definition on quantum analogue. We present some properties and obtain formulaes of ̅q-Laplace transform with its aplications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new transformations for Horn's double hypergeometric functions G1,G2 and G3. Also deduce new integral representations of Euler-type involving these functions in terms of hypergeometric functions of two variables. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to use the newly proposed $(p,q;l)$-extended beta function to introduce the $(p,q;l)$-extended $τ$-Gauss hypergeometric and the $(p,q;l)$-extended $τ$-confluent hypergeometric functions with some of their properties, such as the Laplace-type and the Euler-type integral formulas. Another is to apply them to fractional kinetic equations that appear in astrophysics and physics using the Laplace transform method.

The purpose of this paper is to build some new Hardy-Hilbert-type inequalities with multiparameters and their equivalent forms and variants, which generalize some existing results. Similarly, the corresponding Hardy-Hilbert-type integral inequalities are also given.

Introduction/purpose: The running of the coupling constant in various Quantum Field Theories and a possible behaviour of the beta function are illustrated. Methods: The Callan–Symanzik equation is used for the study of the beta function evolution. Results: Different behaviours of the coupling constant for high energies are observed for different theories. The phenomenon of asymptotic freedom is of particular interest. Conclusions: Quantum Electrodynamics (QED) and Quantum Chromodinamics (QCD) coupling constants have completely different behaviours in the regime of high energies. While the first one diverges for finite energies, the latter one tends to zero as energy increases. This QCD phenomenon is called asymptotic freedom.

Various integral representations of hypergeometric functions have been introduced and investigated due to their important applications in divers fields. In this article, we define some new Euler-type integral representations for the Horn's functions of two variables G1, G2, G3 and H1.

In this paper, we recall some basic facts and preliminaries related to fractional calculus by making a literature review for its definitions and properties. This would provide sufficient knowledge about this important topic. Several lemmas and theorems are shown in detail for completeness. [ABSTRACT FROM AUTHOR]

We evaluate, in closed forms, twelve infinite series that contain a free variable x and binomial coefficients ( 3 n + ϵ n) in the denominators. Several summation formulae resembling Gosper's sum are derived as consequences. [ABSTRACT FROM AUTHOR]

Investigation of coefficient problems is of great interest among function theorists, The nature of coefficients of series functions go along way to determine their properties. In this work therefore, the authors introduce a new subclass of univalent functions denoted by Aω(α) and establish its coefficient estimate. Furthermore, class Aω(α) involving beta function B(m, n) is discussed. [ABSTRACT FROM AUTHOR]

Numerous series representations for various special functions and mathematical constants have been developed by many authors. The aim of this article is to establish two parameterized series representations for the digamma function that seem interesting due to their independence from the given parameters. Among many particular cases of our two main findings, some are covered in the examples. [ABSTRACT FROM AUTHOR]

Dynamic Multi-Objective Optimization Problems (DMOPs) and Many-Objective Optimization Problems (MaOPs) are two classes of the optimization field that have potential applications in engineering. Modified Multi-Objective Evolutionary Algorithms hybrid approaches seem to be suitable to effectively deal with such problems. However, the standard Crow Search Algorithm has not been considered for either DMOPs or MaOPs to date. This paper proposes a Distributed Bi-behaviors Crow Search Algorithm (DB-CSA) with two different mechanisms, one corresponding to the search behavior and another to the exploitative behavior with a dynamic switch mechanism. The bi-behaviors CSA chasing profile is defined based on a large Gaussian-like Beta-1 function, which ensures diversity enhancement, while the narrow Gaussian Beta-2 function is used to improve the solution tuning and convergence behavior. Two variants of the proposed DB-CSA approach are developed: the first variant is used to solve a set of MaOPs with 2, 3, 5, 7, 8, 10,15 objectives, and the second aims to solve several types of DMOPs with different time-varying Pareto optimal sets and a Pareto optimal front. The second variant of DB-CSA algorithm (DB-CSA-II) is proposed to solve DMOPs, including a dynamic optimization process to effectively detect and react to the dynamic change. The Inverted General Distance, the Mean Inverted General Distance and the Hypervolume Difference are the main measurement metrics used to compare the DB-CSA approach to the state-of-the-art MOEAs. The Taguchi method has been used to manage the meta-parameters of the DB-CSA algorithm. All quantitative results are analyzed using the non-parametric Wilcoxon signed rank test with 0.05 significance level, which validated the efficiency of the proposed method for solving 44 test beds (21 DMOPs and 23 MaOPS). [ABSTRACT FROM AUTHOR]

In the present work some integral inequalities which involve the Beta function for stochastic processes whose absolute values posses the property of convexity, or P−convexity or s−convexity in the second sense are established. In the same way some others integral inequalities for stochastic processes whose k−th powers of its absolute values posses these kind of generalized convexity are established making use of the H¨older’s inequality and power mean inequality. [ABSTRACT FROM AUTHOR]

Abstract The beta and gamma functions have recently seen several developments and various extensions because of their nice properties and interesting applications. The contribution of this paper falls within this framework. After introducing a generalized gamma function and two generalized beta functions in several variables, we investigate some inequalities involving these generalized functions.

Bessel function has a significant role in fractional calculus having immense applications in physical and theoretical approach. Present work aims to introduce fractional integral operators in which generalized multi-index Bessel function as a kernel, and develop some important special cases which are connected with fractional operators in fractional calculus. Here, we construct important links to familiar findings from some individual occurrence with our key outcomes.

Abstract In this work, we construct the genuine Durrmeyer–Stancu type operators depending on parameter α in [ 0 , 1 ] $[0,1]$ as well as ρ > 0 $\rho >0$ and study some useful basic properties of the operators. We also obtain Grüss–Voronovskaja and quantitative Voronovskaja types approximation theorems for the aforesaid operators. Further, we present numerical and geometrical approaches to illustrate the significance of our new operators.

A new definition of the incomplete beta function as a distribution-valued meromorphic function is given and the finite parts of it and of its partial derivatives at the singular values are calculated and compared with formulas in the literature. [ABSTRACT FROM AUTHOR]

The four-parameter kappa distribution (K4D) is a generalized form of some commonly used distributions such as generalized logistic, generalized Pareto, generalized Gumbel, and generalized extreme value (GEV) distributions. Owing to its flexibility, the K4D is widely applied in modeling in several fields such as hydrology and climatic change. For the estimation of the four parameters, the maximum likelihood approach and the method of L-moments are usually employed. The L-moment estimator (LME) method works well for some parameter spaces, with up to a moderate sample size, but it is sometimes not feasible in terms of computing the appropriate estimates. Meanwhile, using the maximum likelihood estimator (MLE) with small sample sizes shows substantially poor performance in terms of a large variance of the estimator. We therefore propose a maximum penalized likelihood estimation (MPLE) of K4D by adjusting the existing penalty functions that restrict the parameter space. Eighteen combinations of penalties for two shape parameters are considered and compared. The MPLE retains modeling flexibility and large sample optimality while also improving on small sample properties. The properties of the proposed estimator are verified through a Monte Carlo simulation, and an application case is demonstrated taking Thailand's annual maximum temperature data. [ABSTRACT FROM AUTHOR]

In this research paper, we generalized the results of Ignacy in the multivariate case in order to characterize the complex Wishart distribution. [ABSTRACT FROM AUTHOR]

The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific members of the matrix polynomial family are shown. [ABSTRACT FROM AUTHOR]

Abstract In this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.

This paper studies a numerical approach based on generalized fractional-order Chebyshev wavelets for solving distributed-order fractional optimal control problems (DO-FOCPs). The exact value of the Riemann–Liouville fractional integral operator of the given wavelets is computed by applying regularized beta function. The exact formula and collocation method are applied to transform the DO-FOCP to a new optimization problem. This new problem can be solved by the existing methods. Four examples are given to show the advantage of this method in comparison with the existing methods in the literature. [ABSTRACT FROM AUTHOR]

This article deals with inferences on stress strength reliability in a multicomponent system for Type I generalized half-logistic distribution. It is assumed that the strength and stress components are independently distributed. In this work, we develop some statistical properties of the type I generalized half-logistic distribution. Furthermore, the expression for stress strength reliability for a multicomponent setup was obtained and studied. Two methods to estimate the multicomponent stress-strength reliability - maximum likelihood and Bayesian estimation were employed. The Bayes estimates of the multicomponent stress strength reliability are obtained under squared error loss function and using gamma priors for the parameters. Simulation studies were conducted to assess the efficiency of the methods. The importance of this model was studied by applying it to a real life data set. [ABSTRACT FROM AUTHOR]

This paper proposes a new numerical method for solving fractional optimal control problems (FOCPs). The method is based on generalised fractional-order Chebyshev wavelets (GFOCW). The exact value of the Riemann–Liouville fractional integral operator of the GFOCW is given by applying the incomplete beta function. By using the properties of GFOCW and the collocation method, the FOCP is reduced to a parameter optimisation problem. The last problem is solved by known algorithms. Six numerical examples are given. One of them is an application example in a cancer model. Through these numerical examples, we will show that for some cases of our examples, we will get the exact solutions. These solutions were not obtained previously in the literature. In addition, our method gives more accurate results in comparison with the existing methods. [ABSTRACT FROM AUTHOR]

A new alternative numerical method for solving variable‐order fractional optimal control problems (VO‐FOCPs) is introduced. We use fractional‐order Chebyshev wavelets for solving these VO‐FOCPs. By applying the regularized beta functions, an exact value for the fractional integration of the given wavelets is provided. By applying this formula and the given wavelets, we reduce the given VO‐FOCP to a system of algebraic equations which can be solved by using known methods. For the function approximation of our method, we also give the convergence analysis. By using several numerical examples, we show that this method is more accurate than the other existing methods in the literature. [ABSTRACT FROM AUTHOR]

The expression of genes that control germination-related processes in corn (Zea mays L.) is influenced by environmental factors. Germination of seeds may be facilitated by hormonal priming. The purpose of this investigation was to quantify the effects of different germination temperatures [(5, 10, 15, 20, 25, 30, 35, and 40°C), NaCl-induced stress (0, −0.4, −0.8, and −1.2 MPa), and priming solutions (control, hydropriming, abscisic acid (ABA), gibberellic acid (GA), and salicylic acid (SA)] (Experiment 1). Effects of germination temperatures, PEG 6000-induced stress (0, −0.4, −0.8, and −1.2 MPa), and priming solutions were also evaluated separately (Experiment 2). In both cases, a completely randomized design with four replications was used. Increasing temperatures from 5 to 25°C gradually improved germination percentage and rate, whereas temperatures > 25°C decreased these indices. After imposing drought (PEG 6000-induced stress) or salinity (NaCl-induced stress) treatments, hormonal priming caused germination to occur at a lower base temperature, compared with the non-priming treatment. However, the effect of hormonal priming was dependent on temperature. At sub-optimal temperatures (< 25°C), the highest germination percentage and rate were recorded after GA priming. At above-optimal temperatures (> 25°C), ABA priming resulted in the highest germination percentage and rate. Moreover, hydrothermal time constant decreased in hormone-treated seeds. Based on coefficient of determination (R2) and root mean square error (RMSE), a dent-like model predicted cardinal temperatures more accurately than a beta model did. Generally, GA-, SA-, and ABA-priming were recommended under sub-optimal, optimal, and above-optimal temperatures, respectively. [ABSTRACT FROM AUTHOR]

The main object of this paper is to introduce a new extension of beta function involving generalized Mittag-leffler function and study its important properties, like integral representation, summation formula, derivative formula, beta distribution, transform formula. Using this definition, we introduce new extended hypergeometric and con uent hypergeometric function. [ABSTRACT FROM AUTHOR]