Search

Measure-valued structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure-valued structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary. [ABSTRACT FROM AUTHOR]

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions. [ABSTRACT FROM AUTHOR]

The purpose of this note is to specify for lattice homomorphisms Riesz type representation theorems obtained by Wright and Coquand for positive operators. We prove that every lattice homomorphism acting from a vector lattice of continuous functions on a Hausdorff compact topological space with values in a Dedekind complete or Dedekind σ -complete vector lattice admits an integral representation by a quasi-regular vector measure that is a σ -continuous Boolean homomorphism. We use an operator approach to obtain suitable vector measures to represent lattice homomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic properties of slowly varying functions of one real variable in the sense of Karamata. We establish analogs of fundamental theorems on uniform convergence and integral representation for slowly varying functions with a remainder depending on the types of remainder. We also prove several important theorems on the asymptotic representation of integrals of Karamata functions. Under certain conditions, we observe a "narrowing" of classes of slowly varying functions concerning the types of remainder. At the end of the paper, we discuss the possibilities of the application of slowly varying functions in the theory of stochastic branching systems. In particular, under the condition of the finiteness of the moment of the type E x ln x for the particle transformation intensity, it is established that the property of slow variation with a remainder is implicitly present in the asymptotic structure of a non-critical Markov branching random system. [ABSTRACT FROM AUTHOR]

The finite Hilbert transform T$T$, when acting in the classical Zygmund space LlogL$L\textnormal {log} L$ (over (−1,1)$(-1,1)$), was intensively studied in [8]. In this note, an integral representation of T$T$ is established via the L1(−1,1)$L^1(-1,1)$‐valued measure mL1:A↦T(χA)$m_{L^1}: A\mapsto T(\chi _A)$ for each Borel set A⊆(−1,1)$A\subseteq (-1,1)$. This integral representation, together with various non‐trivial properties of mL1$m_{L^1}$, allows the use of measure theoretic methods (not available in [8]) to establish new properties of T$T$. For instance, as an operator between Banach function spaces T$T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for T$T$ plays a crucial role. [ABSTRACT FROM AUTHOR]

In this article, we generalized Mittag‐Leffler‐type functions F¯A(3),F¯B(3),F¯C(3)$$ {\overline{F}}_A^{(3)},{\overline{F}}_B^{(3)},{\overline{F}}_C^{(3)} $$, and F¯D(3)$$ {\overline{F}}_D^{(3)} $$, which correspond, respectively, to the familiar Lauricella hypergeometric functions FA(3),FB(3),FC(3)$$ {F}_A^{(3)},{F}_B^{(3)},{F}_C^{(3)} $$, and FD(3)$$ {F}_D^{(3)} $$ of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function F¯D(3)$$ {\overline{F}}_D^{(3)} $$, which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them. [ABSTRACT FROM AUTHOR]

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

We establish the subadditivity of strongly operator convex functions on (0 , ∞) and (− ∞ , 0). By utilizing the properties of strongly operator convex functions, we derive the subadditivity property of operator monotone functions on (− ∞ , 0). We introduce new operator inequalities involving strongly operator convex functions and weighted operator means. In addition, we explore the relationship between strongly operator convex and Kwong functions on (0 , ∞). Moreover, we study strongly operator convex functions on (a , ∞) with − ∞ < a and on the left half-line (− ∞ , b) with b < ∞. We demonstrate that any nonconstant strongly operator convex function on (a , ∞) is strictly operator decreasing, and any nonconstant strongly operator convex function on (− ∞ , b) is strictly operator monotone. Consequently, for a strongly operator convex function g on (a , ∞) or (− ∞ , b) , we provide lower bounds for | g (A) − g (B) | whenever A − B > 0. [ABSTRACT FROM AUTHOR]

In this article, we generalized Mittag‐Leffler‐type functions F¯A(3),F¯B(3),F¯C(3)$$ {\overline{F}}_A&#x0005E;{(3)},{\overline{F}}_B&#x0005E;{(3)},{\overline{F}}_C&#x0005E;{(3)} $$, and F¯D(3)$$ {\overline{F}}_D&#x0005E;{(3)} $$, which correspond, respectively, to the familiar Lauricella hypergeometric functions FA(3),FB(3),FC(3)$$ {F}_A&#x0005E;{(3)},{F}_B&#x0005E;{(3)},{F}_C&#x0005E;{(3)} $$, and FD(3)$$ {F}_D&#x0005E;{(3)} $$ of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function F¯D(3)$$ {\overline{F}}_D&#x0005E;{(3)} $$, which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them. [ABSTRACT FROM AUTHOR]

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev's integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard's numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. [ABSTRACT FROM AUTHOR]

We prove the existence of solutions to the boundary value problem for a system of five nonlinear second order partial differential equations, with given nonlinear boundary conditions, describing the equilibrium state of elastic shallow inhomogeneous anisotropic shells with free edges within the framework of Timoshenko's shear model, referred to arbitrary curvilinear coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle. [ABSTRACT FROM AUTHOR]

The classical growth indicators of entire and meromorphic functions are order and type, which are generalized by several authors during the past decades. Chyzhykov et al. have first introduced the generalized growth scale, namely the φ-order (see [3]) taking φ as an increasing unbounded function. But, Heittokangas et al. [5] have introduced another new concept of φ-order of entire and meromorphic functions considering φ as subadditive function. Later, Bela¨ıdi et al. [1] have extended the above ideas and have introduced the definition of (α, β, γ)-order of entire and meromorphic functions, where α ∈ L¹-class, β ∈ L2-class, γ ∈ L3-class. In this paper, our motive is to develop the integral representations of (α, β, γ)- order and (α, β, γ)-lower order of a meromorphic function. We also investigate their equivalence relation under some certain conditions. [ABSTRACT FROM AUTHOR]

We give a complete characterization of all real-valued functions on the unit circle \(S^1\) that can be represented by integrating the spherical distance on \(S^1\) with respect to a signed measure or a probability measure. [ABSTRACT FROM AUTHOR]

In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered. [ABSTRACT FROM AUTHOR]

We prove an integral representation result for a class of variational functionals appearing in the framework of hierarchical systems of structured deformations via a global method for relaxation. Some applications to specific relaxation problems are also provided. [ABSTRACT FROM AUTHOR]

In this article, it is constructed special solutions for a certain class of degenerating differential equations of hyperbolic type of a second order. Linearly independent solutions consisting of hypergeometric functions are found for the given partial differential equation of hyperbolic type. These self-similar solutions are expressed in terms of Gauss hypergeometric functions. [ABSTRACT FROM AUTHOR]

Hierarchical (first-order) structured deformations are studied from the variational point of view. The main contributions of the present research are the first steps, at the theoretical level, to establish a variational framework to minimize mechanically relevant energies defined on hierarchical structured deformations. Two results are obtained here: (i) an approximation theorem and (ii) the assignment of an energy to a hierarchical structured deformation by means of an iterative procedure. This has the effect of validating the proposal made in Deseri and Owen (J. Elast. 135:149–182, 2019) to study deformations admitting slips and separations at multiple submacroscopic levels. An explicit example is provided to illustrate the behavior of the proposed iterative procedure and relevant directions for future research are highlighted. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish the monotonicity results of the ratios between normalized tails of the Maclaurin power series expansions of the sine and cosine functions and restate them in terms of the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]

Herein, the Laplace transform representations for functions of weighted holomorphic Bergman spaces on the tube domains are developed. Then a weighted version of the edge-of-the-wedge theorem is derived as a byproduct of the main results. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

The realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated K-moment problem where K is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited by Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253–1281], where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained by Curto, Ghasemi, Infusino, and Kuhlmann [J. Operator Theory 90 (2023), pp. 223–261] for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results of Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253–1281], allowing to finally embed them in the above-mentioned unified framework for the infinite-dimensional truncated moment problem. [ABSTRACT FROM AUTHOR]

Integral representations for a complete set of linearly independent products of two solutions of the Airy equation whose arguments differ by z 0 are obtained using the Laplace contour integral method. This generalizes similar integral representations for the case z 0 = 0 obtained by Reid. The relation to other previous results is discussed. The results are used to obtain the outgoing-wave Green’s function for an electron in a static electric field in a closed analytic form. [ABSTRACT FROM AUTHOR]

The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances. [ABSTRACT FROM AUTHOR]

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions.

In this paper, we study the asymptotic properties of slowly varying functions of one real variable in the sense of Karamata. We establish analogs of fundamental theorems on uniform convergence and integral representation for slowly varying functions with a remainder depending on the types of remainder. We also prove several important theorems on the asymptotic representation of integrals of Karamata functions. Under certain conditions, we observe a “narrowing” of classes of slowly varying functions concerning the types of remainder. At the end of the paper, we discuss the possibilities of the application of slowly varying functions in the theory of stochastic branching systems. In particular, under the condition of the finiteness of the moment of the type Exlnx for the particle transformation intensity, it is established that the property of slow variation with a remainder is implicitly present in the asymptotic structure of a non-critical Markov branching random system.

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included. [ABSTRACT FROM AUTHOR]

We extend previous research to derive three additional M-1-dimensional integral representations over the interval [ 0 , 1 ] . The prior version covered the interval [ 0 , ∞ ] . This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) | x 1 − x 2 | = x 1 2 − 2 x 1 x 2 cos θ + x 2 2 to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval [ 0 , 1 ] are numerically stable, as was the prior version, having integrals running over the interval [ 0 , ∞ ] , and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over [ 0 , 1 ] than over [ 0 , ∞ ] . In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form (a x 2 + b x + c) / x , for which a single tabled integral exists for the integrals from running over the interval [ 0 , ∞ ] of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval [ 0 , 1 ] . Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over [ 0 , 1 ] . [ABSTRACT FROM AUTHOR]

The well-posedness of an initial–boundary value problem for a model parabolic equation with two power-law degeneration lines is studied. Two initial–boundary value problems depending on the degeneracy exponents are stated, and uniqueness and existence theorems are proved. The solutions of these problems are constructed in closed form. [ABSTRACT FROM AUTHOR]

The concept of covariogram is extended from bounded convex bodies in to the entire space by obtaining integral representations for the distribution and probability density functions of the Euclidean distance between two -dimensional Gaussian points that have correlated coordinates governed by a covariance matrix. When , a closed-form expression for the density function is obtained. Precise bounds for the moments of the considered distance are found in terms of the extreme eigenvalues of the covariance matrix. [ABSTRACT FROM AUTHOR]

The solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates, is studied. The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in a Sobolev space, with the solvability of this equation being established using the contraction mapping principle. [ABSTRACT FROM AUTHOR]

In complex analysis, order and type are classical growth indicators of entire and meromorphic functions. During the past decades, several authors have made the close investigations on the properties of entire and meromorphic functions in different directions using the concepts of order, iterated p-order [8, 11], (p, q)-th order [6, 7], (p, q)-φ order [10] and achieved many valuable results. But in [3], Chyzhykov et al. showed that both definitions of iterated porder and (p, q)-th order have the disadvantage that they do not cover arbitrary growth (see [3, Example 1.4]). They used more general scale, called the φ-order (see [3]). On the other hand, Heittokangas et al. [4] have introduced another new concept of φ-order of entire and meromorphic functions considering φ as subadditive function. Considering all these aspects, Belaıdi et al. [1, 2] have extended the above ideas and have introduced the definitions of (α, β, γ)- order and (α, β, γ)-type of entire and meromorphic functions. In this paper, we establish the integral representations of (α, β, γ)-type and (α, β, γ)-weak type of a meromorphic function. We also investigate their equivalence relation under some certain conditions. [ABSTRACT FROM AUTHOR]

In the present paper, we establish the incomplete exponential type (IEF) of $ R $-matrix functions and identify some properties of the incomplete exponential matrix functions including integral representation, some derivative formula and generating functions of the incomplete exponential of $ R $-matrix functions. Finally, special cases of the presented results are pointed out.

In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.

Lauricella, G. in 1893 defined four multidimensional hypergeometric functions F A , F B , F C and F D . These functions depended on three variables but were later generalized to many variables. Lauricella's functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella's function F A (n) , the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered F A (n) function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used. [ABSTRACT FROM AUTHOR]

This work aims to introduce κ-Appell hypergeometric matrix functions F1,κ,F2,κ,F3,κ and F4,κ, where κ ∈ N. New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with κ-Appell hypergeometric matrix functions F1,κ,F2,κ are derived. [ABSTRACT FROM AUTHOR]

In the present paper, we establish the incomplete exponential type (IEF) of R-matrix functions and identify some properties of the incomplete exponential matrix functions including integral representation, some derivative formula and generating functions of the incomplete exponential of R-matrix functions. Finally, special cases of the presented results are pointed out. [ABSTRACT FROM AUTHOR]

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method. [ABSTRACT FROM AUTHOR]

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals.

This work aims to introduce [Formula: see text]-Appell hypergeometric matrix functions [Formula: see text] and [Formula: see text], where [Formula: see text]. New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with [Formula: see text]-Appell hypergeometric matrix functions [Formula: see text] are derived.

In this paper, we derive some classical and fractional properties of the r R s matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given. [ABSTRACT FROM AUTHOR]

Given an open and bounded set Ω ⊆ ℝ n and a family 퐗 = (X 1 , ... , X m) of Lipschitz vector fields on Ω, with m ≤ n , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ (u , A) = ∫ A f ⁢ (x , u ⁢ (x) , X ⁢ u ⁢ (x)) ⁢ 푑 x , with f being a Carathéodory integrand. [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, we give another method to calculate the values of multiple Hurwitz zeta function at non-positive integers by means of Raabe’s formula and the Bernoulli numbers and we simplify these values by symbolic computation techniques.

In this paper the product of classical Riemann zeta function ζ (s) and fractional hypergeometric zeta functions ζ a (s) is examined. As a result of this product a new integral representation of the square of the Riemann zeta function is obtained. We use this integral representation to show that some special Mellin transforms which are naturally related to the fractional hypergeometric zeta functions have zero free regions in the specified right half of the complex plane. Finally we show that the fractional hypergeometric zeta functions ζ a (s) are zero free for some specified values of a. [ABSTRACT FROM AUTHOR]