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This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable. [ABSTRACT FROM AUTHOR]

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group SO (2 , 2) . Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group SO (2 , 2) is connected to symmetry in several significant ways, especially in mathematical physics and geometry. [ABSTRACT FROM AUTHOR]

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case G = GL r we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group U q (gl ^ (r | 1)) , which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials. [ABSTRACT FROM AUTHOR]

This paper is set to analytically describe properties of the hyperbolic distribution. This law, along with the variance-gamma distribution, is one of the most popular normal mean–variance mixtures from the point of view of various applications. We have found closed form expressions for the cumulative distribution and partial-moment-generating functions of the hyperbolic distribution. The obtained formulas use the values of the Humbert confluent hypergeometric and Whittaker special functions. The results are applied to the problem of European option pricing in the related Lévy model of financial market. The research demonstrates that the discussed normal mean–variance mixture is analytically tractable.

In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform. [ABSTRACT FROM AUTHOR]

This study is mainly focused on the behavior change of shear waves. These waves propagate through an inhomogeneous sandwiched water-saturated porous layer in between inhomogeneous elastic layer and inhomogeneous sandy half space under the assumption of non-local theory. Different types of inhomogeneities based on the depth of the layers have been considered in terms of elastic moduli and densities of the medium. Due to the availability of fluid in the second layer, we consider loose bonding at the interfaces in between these three layers, as fluid movement through the interfaces may be possible. A complex dispersion equation has been derived with respect to the boundary conditions. Real and imaginary parts of the dispersion equation provide us with the phase velocity of shear wave and dissipation due to loose bonding at the interfaces, respectively. This dispersion equation has been solved numerically with the help of Newton Raphson method. For validation of the present problem, the numerical solution of dispersion equation has been depicted here graphically in terms of phase velocity c c 2 of shear wave vs. non-dimensional wave number k H 1 (where k is the wave number and H 1 is the thickness of the second porous layer). Also, we discuss about the impact of depth ratios, inhomogeneity of the medium, non-local parameters, loose bonding, and porosity of the sandwiched layer on the propagation of the shear wave. [ABSTRACT FROM AUTHOR]

In this paper, by using an integral representation of Ismail and Kelker for the quotient of Tricomi hypergeometric functions, we investigate the infinite divisibility and self-decomposability of the recently defined four-parameter lifetime Whittaker distribution, which is a natural extension of the classical gamma, exponential, chi-square, generalized Lindley, Lindley, beta prime, and Lomax distributions. We also show that the Whittaker distribution belongs to the class of hyperbolically completely monotone distributions and generalized gamma convolutions, and it is a super-Gaussian distribution. By using some results for the moments of the Whittaker distribution, we also deduce some Turán type inequalities for the Whittaker functions of the second kind and as an application we show that the effective variance of the Whittaker distribution is bounded from below. [ABSTRACT FROM AUTHOR]

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group SO(2,2). Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group SO(2,2) is connected to symmetry in several significant ways, especially in mathematical physics and geometry.

In this paper, we study the fractional-order system in the sense of ϑ -Hilfer fractional stochastic evolution equations driven by fractional Brownian motion. Applying the fixed point technique, we prove that there exists a mild solution for the problem and introduce a new type of stability. Finally, we present two examples to demonstrate how the obtained results might be applied. [ABSTRACT FROM AUTHOR]

Image forgery is a crucial part of the transmission of misinfor)mation, which may be illegal in some jurisdictions. The powerful image editing software has made it nearly impossible to detect altered images with the naked eye. Images must be protected against attempts to manipulate them. Image authentication methods have gained popularity because of their use in multimedia and multimedia networking applications. Attempts were made to address the consequences of image forgeries by creating algorithms for identifying altered images. Because image tampering detection targets processing techniques such as object removal or addition, identifying altered images remains a major challenge in research. In this study, a novel image texture feature extraction model based on the generalized k-symbol Whittaker function (GKSWF) is proposed for better image forgery detection. The proposed method is divided into two stages. The first stage involves feature extraction using the proposed GKSWF model, followed by classification using the “support vector machine” (SVM) to distinguish between authentic and manipulated images. Each extracted feature from an input image is saved in the features database for use in image splicing detection. The proposed GKSWF as a feature extraction model is intended to extract clues of tam)pering texture details based on the probability of image pixel. When tested on publicly available image dataset “CASIA” v2.0 (Chinese Academy of Sciences, Institute of Automation), the proposed model had a 98.60% accuracy rate on the YCbCr (luminance (Y), chroma blue (Cb) and chroma red (Cr)) color spaces in image block size of 8 × 8 pixels. The proposed image authentication model shows great accuracy with a relatively modest dimension feature size, supporting the benefit of utilizing the k-symbol Whittaker function in image authentication algorithms. [ABSTRACT FROM AUTHOR]

In this paper, using the generalized Whittaker translation established by Sousa et al., we shall prove a generalization version of Titchmarsh's theorem for the modified Whittaker transform F W for functions satisfying the Whittaker-Lipschitz condition in the appropriate space L 2 (m). [ABSTRACT FROM AUTHOR]

Introduction/purpose: The aim of this article is to establish integral transforms of the generalized Lommel-Wright function. Methods: These transforms are expressed in terms of the Wright Hypergeometric function. Results: Integrals involving the trigonometric, generalized Bessel function and the Struve functions are obtained. Conclusions: Various interesting transforms as the consequence of this method are obtained.

In this paper, we introduce a generalized form of Whittaker function with the help of generalized conuent k-hypergeometric function. We establish several interesting properties of the Whittaker k-function such as its integral representations, derivative, Laplace transform and Hankel transform. Further, we investigate the Riemann-Liouville fractional integral and k-Riemann-Liouville fractional integral of Whittaker k-function. Some intriguing particular cases of the main results are also mentioned. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a general family of distributions based on Whittaker function. The properties of obtained distributions, moments, ordering, percentiles, and unimodality are studied. The distributions' parameters are estimated using methods of moments and maximum likelihood. Furthermore, a generalization of Whittaker distribution that contains a wider class of distributions is developed. Validation of the obtained results is applied to real life data containing four data sets. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The aim of this article is to establish integral transforms of the generalized Lommel-Wright function. Methods: These transforms are expressed in terms of the Wright Hypergeometric function. Results: Integrals involving the trigonometric, generalized Bessel function and the Struve functions are obtained. Conclusions: Various interesting transforms as the consequence of this method are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we investigate integral transforms involving the product of the Bessel function of the first kind and orthogonal polynomials with the weight e - p ρ 2 + 2 q ρ . These integral transforms are expressed in terms of a series of Humbert function Ψ 2 or Lauricella function F A (3) of three variables. With the help of our main results, certain special cases are also examined. The application of the main results is observed to study an intensity phenomena known as diffraction of Laguerre-Gauss and Laguerre-Bessel-Gausian beams by a curved fork-shaped grating. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of material strength due to the traction and propagation of torsional surface waves. Dynamic behavior of material strength depends on non-homogeneity, aeolotropy, irregularity, porosity, and internal prestress in an elastic body. The geometry of the model is constructed by a local-elastic, highly non-homogeneous finite thickness layer over a trigonometric variation of non-homogeneous aeolotropic porous half-space. The surface boundaries of both the free surface and interface are considered corrugated boundaries. The separation variable technique is used to solve the equation of motions (hyperbolic type PDEs), and the Bessel function is adopted in the displacement components of torsional waves for both the media. Because of highly non-homogeneity in the upper layer, the equation of motion comprises a special form of differential equation called Whittaker differential equation, and the solution (as displacement) of the equation is obtained in the form of Whittaker functions. The quantitative prediction of a differential equation is one of the significant tasks in the study of wave propagation, thus, the dispersion equation of the torsional surface waves is introduced analytically with the help of traction-free and continuity conditions. The validity of the final dispersion equation is described using several particular cases. To study the intensity of the torsional surface waves in the model, traction components have been computed along the coordinate axes, which helps to detect the material fracture in the medium. MATLAB software is used to study the dynamic behavior of the phase velocity, under the influence of prestressing irregular boundary surface, non-homogeneity, and porosity parameters in the media. By the numerical simulations, some marvelous changes in the propagation of the torsional surface waves due to the irregularity at the interface and free surface are obtained. The work may be useful to study the behavior of torsional surface waves in the micro-structure material and smart material composite structures used in the aerospace industry and seismology. [ABSTRACT FROM AUTHOR]

In this article, we give an integral representation of the resolvent kernel on H-type groups, then we derive an integral representation of Kaplan's fundamental solution on this groups. Also we obtain the Green kernel for fractional powers of its sub-Laplacian. [ABSTRACT FROM AUTHOR]

For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.

We give an integral representation of the fundamental solution of the heat and resolvent equations associated to the Kohn's Laplacian on the octonionic Heisenberg group. Moreover, the fundamental solution for the octonionic Heisenberg Laplacian is found. [ABSTRACT FROM AUTHOR]

We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah [3, Theorem 1.1]. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface. [ABSTRACT FROM AUTHOR]

In this paper, a transformation is applied to Bessel-Circular-Gaussian beam to generate a new laser beam called Exton-Gaussian beams which have applications to wide areas of optics. For this, a novel integral transform involving the product of the Whittaker function and two Bessel functions of the first kind; the first with a linear argument and the second, which is a modified Bessel function, with a quadratic argument is derived whose solution is expressed in terms of the Exton's function X8 and the Lauricella's function F(3)A . Some special cases of the main integral are illustrated by being specific on parameters. The analytical expression of the beam is utilized to study the propagation behavior. It is shown that the beam is composed of lobes and a dark center, potentially capable of trapping atoms in it. Such a behavior is important in physics and will be very useful for trapping atoms in its dark center and could be exploited as optical tweezers. [ABSTRACT FROM AUTHOR]

Aim: The study is to investigate the impact of torsional surface wave in dry sand-filled desert whose upper surface is irregular due to transverse sand dunes. Method and Consideration: In the desert, the rigidity and density of the granular soil are presumed to vary in an exponential form with the depth of it. The free surface of the desert has been contemplated as I) traction frees, II) rigid boundary. The four scattering equations for the cases are obtained in a closed form by means of separation variables method in the presence of Whittaker differential equation. Some special cases are derived, and their existence has been discussed. The influences of inhomogeneity factor, irregularity of dunes and sandy parameter on the phase velocity have also been studied graphically in the cases. Purpose: The result is useful to realize geo-dynamics of the sandy desert, earthquake engineering, civil engineering construction of road and building on the sandy medium, oil exploration, artificial exploration, etc. [ABSTRACT FROM AUTHOR]

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer's confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results. [ABSTRACT FROM AUTHOR]

This article presents a comprehensive study of torsional surface wave propagation in a piezoelectric fiber-reinforced composite (PFRC) layer bonded to functionally graded (FG) half-space considering electrically short and open boundary conditions. The Strength of Materials (SM) and Rule of Mixtures (ROM) approaches have been adopted to determine the effective properties of the PFRC. The non-ideal mechanical and electrical interfacial boundary has been considered for the problem which appears in various realistic scenarios. For the torsional surface wave propagation through PFRC bonded to FG half-space, the dispersion equation has been derived analytically using Whittaker function. The obtained results are validated and matched with existing results by considering special cases. The variation of phase velocity due to the influence of fiber volume fraction, interfacial imperfection parameters and functionally grading parameters have been analyzed and illustrated by means of graphs. Furthermore, the distribution of the field variables, stress and electric displacements across the composite are presented through surface plots. The model holds potential applications in designing and manufacturing transducers, actuators and sensors. The use of PFRC can be found in a wide range of cutting-edge technology, including vibration-canceling gadgets, energy harvesters, and pressure sensors. • The analytical study of the torsional surface wave propagation in piezo-electric fiber-reinforced composite (PFRC) structure is investigated. • Strength of material (SM) and Rule of mixtures (RoM) are adopted todetermine the effective coefficients of the micro-mechanical modeling ofPFRC structures. • The effect of fibre-matrix volume fraction, functionally grading parameters and imperfect boundary on the propagation of the torsional wave in PFRC structure arediscussed analytically as well as numerically. • The 2D and 3D plots of mode shapes of field variables for torsional wave propagation are presented graphically. [ABSTRACT FROM AUTHOR]

Abstract By using the method of weight function, the technique of real analysis, and the theory of special functions, a multi-parameter Hilbert-type integral inequality and its equivalent form are established, and their constant factors are proved to be the best possible. The expressions of operator with norm are given. As an application, relevant results in the references and some new inequalities are obtained by assigning some parameter values.

Let V be a rational quadratic space of signature (m; 2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m + 1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimedian and non-archimedian places. [ABSTRACT FROM AUTHOR]

Discrete analogs of the index Whittaker transform are introduced and investigated. It involves series and integrals with respect to a second parameter of the Whittaker function W μ , in (x) , x > 0 , μ ∈ R , n ∈ N , i is the imaginary unit. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a general family of distributions based on Whittaker function. The properties of obtained distributions, moments, ordering, percentiles, and unimodality are studied. The distributions’ parameters are estimated using methods of moments and maximum likelihood. Furthermore, a generalization of Whittaker distribution that contains a wider class of distributions is developed. Validation of the obtained results is applied to real life data containing four data sets.

In this paper, we define a new extension of Srivastava's triple hypergeometric functions by using a new extension of Pochhammer's symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results. [ABSTRACT FROM AUTHOR]

We give an integral representation of the fundamental solution of the heat and resolvent equation associated to the Casimir–Laplace operator for the quaternionic Heisenberg group. As an application we obtain the corresponding wave kernel and Green function. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new integral representation of the resolvent kernel for the Casimir-Laplacian on the quaternionic Heisenberg group which was obtained in [14] and then find its spectral density. Also we obtain the Green kernel for fractional powers of the Casimir-Laplace operator.

The objective of this paper is to study the effect of loosely bonded interface on torsional surface wave propagation in a fiber reinforced composite medium constrained between dry sandy layer and an anisotropic gravitating poroelastic substrate. All the media are assumed to be under initial stress. The dispersion relation on this proposed multilayer ground structure has been derived in closed form under certain boundary conditions, which contain Whittaker function and its derivative, which is further expanded asymptotically, retaining up to only the linear terms. The numerical solution for the limiting case of torsional surface waves is also discussed. As a special case of the problem, when the entire medium is isotropic and one of the upper layer vanishes and removing the initial stress and gravity, the dispersion relation obtained is in agreement with the classical Love type wave equation. The influence of various technical constants, such as sandy parameter, reinforcement parameter, porosity parameter, Biot's gravity parameter, loosely bonded parameters, initial stress of both the layers and half spaces on the phase velocity of torsional surface wave has been pointed out by means of graphs. [ABSTRACT FROM AUTHOR]

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The problem of constructing asymptotics describing far-field internal gravity waves generated by an oscillating point source of perturbations moving in a vertically semi-infinite stratified layer of variable buoyancy is considered. For a model distribution of the buoyancy frequency, analytical solutions of the main boundary value problem are obtained, which are expressed in terms of Whittaker functions. An integral representation for the Green's function is obtained, and asymptotic solutions are constructed that describe the amplitude-phase characteristics of internal gravity wave fields in a semi-infinite stratified medium with a variable buoyancy frequency far away from the perturbation source. [ABSTRACT FROM AUTHOR]

Highlights • The closed-form symbolic algebras of Whittaker eigenfunctions for a 2D model. • Boundary conditions with arbitrarily prescribed wall temperature or heat flux. • The revealed common features of compressible laminar thermal capillary flows. • The assumption of linear pressure profile is not required like literature. • Validation of the algorithm at low/moderate Reynolds, Mach and Eckert numbers. Abstract Heat transfer in compressible laminar flow in mini-/micro-channels, a classical and general topic in fields of fuel cells, electronics, micro heat exchanger, etc., is revisited. Based on a two-dimensional continuum flow model, analytical solutions of the dimensionless model are achieved in closed-form symbolic algebras of Whittaker eigenfunctions, corresponding to two kinds of boundary conditions with arbitrarily prescribed wall temperature or wall heat flux. As the eigenvalues and eigenfunctions are independent on the dimensionless quantities, which influence the along-the-channel behaviors, the algorithm reveals the common features of compressible laminar thermal flows. The algorithms do not require the assumption of a linear pressure distribution, which is proved to be untenable in some cases (e.g. constant wall heat flux). The algorithms are validated well by the exact (numerical) computations in exemplary cases of both small and moderate Reynolds number, Mach number and Eckert number of air. Although expressed in a series of eigenfunctions, only several terms (sometimes one or two terms) of solutions are required for a practical computation. [ABSTRACT FROM AUTHOR]

In this article, we introduce a further generalizations of the confluent hypergeometric function and Whittaker function by introducing an extra parameter in the extended con uent hypergeometric function dened by Parmar [15]. We also investigate some integral representations, some integral transforms, differential formulas and recurrence relations of these new generalizations

This paper investigates some variant of Whittaker integral operators on a class of square integrable Boehmians. We define convolution products and derive the convolution theorem which substantially satisfy the axioms necessary for generating the Whittaker spaces of Boehmians. Relied on this analysis, we give a definition and properties of the Whittaker integral operator in the class of square integrable Boehmians. The extended Whittaker integral operator, is well-defined, linear and coincides with the classical integral in certain properties.

We obtain a closed-form formula for the quasi-stationary distribution of the classical Shiryaev martingale diffusion considered on the positive half-line [A,+∞) with A > 0 fixed; the state space's left endpoint is assumed to be the killing boundary. The formula is obtained analytically as the solution of the appropriate singular Sturm-Liouville problem; the latter was first considered in section 7.8.2 of [P. Collet, S. Mart'ınez, and J. San Mart'ın, Quasi-Stationary Distributions. Markov Chains, Diffusions and Dynamical Systems, Springer, Heidelberg, 2013] but has heretofore remained unsolved. [ABSTRACT FROM AUTHOR]