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In this paper, the author first prove the theorem on bilateral generating relations for a certain two-variable generalised hypergeometric polynomials by the group theoretic technique introduced by Weisner. It is then shown how the main theorem can be applied to derive a large variety of bilateral generating functions for various special functions as well as for their various generalizations. Some results given by other researchers are thus observed to follow easily as special cases of the theorem proved in this paper. It is worth noting that special functions play role in the design of filters and approximation theory in communication engineering. [ABSTRACT FROM AUTHOR]

The object of this paper is to develop quardratic transformation of κ-hypergeometric differential equation and also develop new forms of quardratic transformation. Further we establish some properties of κ-hypergeometric function with κ-Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

In the present paper, we explore some new formulae of fractional calculus, including the general class of polynomials given by Srivastava [5] and the generalized Hurwitz–Lerch zeta function given by Nadeem et al. [11]. The obtained results are in the form of hypergeometric function, which are made with the help of Hadamard product. We have also derived some important special cases from the main results. [ABSTRACT FROM AUTHOR]

In this work we begin with a brief survey of the generalization of various Gaussian series introduced so far. One dimensional Gaussian series is Hypergeometric functions of several variables arise in quantum field theory as solutions of the Knizhnik-Zamolodchikov equations. These equations can be considered as generalized equations of hypergeometric type, and their solutions admit integral representations that generalize the classical Euler integrals for hypergeometric functions of one variable. Such an approach allows to connect special functions of the hypergeometric type with topical problems in the theory of representations of Lie algebras and quantum groups. [ABSTRACT FROM AUTHOR]

In this paper, we will find expansion formulas for 5 hypergeometric functions of two variables from Horn's list, which includes 34 functions, and for 11 of them, expansion formulas were previously known according to works of Burchnall and Chaundy. To solve the problem, we introduce operators that generalize the well-known Burchnall-Chaundy's operators, study the properties of these new operators, and apply them to finding expansion formulas. At the end of the paper, to give an example, we will show the application of the expansion formula for one of the five functions to determine the order of the singularity at the origin of the fundamental solution used in solving boundary value problem for the multidimensional Helmholtz's equation with one singular coefficient. [ABSTRACT FROM AUTHOR]

Most of the results in special functions have extensive applications in modern engineering problems like wave propagation and image processing. In previous article, we discussed bilateral relations for Bn(α ,β)(x, y, v) with generalized Sylvester polynomial in modified form(MGSP) and quoted few applications as special cases. Now, in this article it has been discussed some more applications for it in continuation to it and these results are general in nature in their manifold. [ABSTRACT FROM AUTHOR]

In this paper, certain properties of newly defined generalized hypergeometric polynomials in two variables{Rn (β ;γ ; x, y)} have been derived, namely, recurrence relations of ascending and descending type, ordinarydifferential equation and linear generating function, which are essential to derive generating relations of various types from the group-theoretic method point of view. Furthermore, Laguerre polynomials of single and two variables, Meixner, Gottlieb and Krawtchouk polynomials are deduced as special cases, which arises in mathematical physics and communication engineering. [ABSTRACT FROM AUTHOR]

A study of few properties of 2-variable generalized hypergeometric polynomial(2VGHP) of the set Rn (β ; γ ; x, y) [1], as sum and product formulae and we derive this polynomial can be represented as some integral representations such as single and double integral representations. Many supplemental applications, as well as known and unknown hypergeometric functions can be obtained by these results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish an expansion formula for the composition of a general class of functions and Srivastava polynomial function, V-function, and H-function by using fractional Fourier transform and also obtain several special results. The expansion formula being of a general nature can be transformed to yield various latest outcomes involving several frequently used functions, viz Bessel function, generalized hypergeometric function, Riemann Zeta function, Mittag-Leffler function, etc. [ABSTRACT FROM AUTHOR]

The intention of the current exploration is to create new extension of generalized Bessel-Maitland function. In addition, investigated several useful properties including integral representation of a new extended function of Laplace Transform, Whittaker Transform and various other transforms in terms of generalized Wright hypergeometric function. Certain belongings of the Riemann-Liouville fractional calculus connected with new extended Bessel-Maitland function are also examined. [ABSTRACT FROM AUTHOR]

The analytical solution of The Klein Gordon equation was solved using the hypergeometry method for Hyperbolic Scarf plus Gendenstein III potential. The Klein Gordon equation for spin symmetry is reduced to the differential of one-dimensional Schrodinger-like equation. The wave function and the corresponding energy eigenvalues equation are simply obtained using variable separation and hypergeometric method. The numerical result of relativistic energy is calculated by Matlab R2013. These results can be applied to determines the thermodynamic's properties of the system such as vibrational mean-energy, vibrational specific heat , and vibrational mean-free energy. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to create ties between different kinds of documents. By adding a certain convolution operator involving hypergeometric functions, subclasses of harmonic univalent functions. In the open unit disk ℒ, we examine certain relations with Goodman-Salagean-Type harmonic univalent functions. [ABSTRACT FROM AUTHOR]

This paper deals with three classes of nonlinear Schrödinger equations (NLSE) having general polynomial nonlinearities. We find travelling wave solutions of soliton type (smooth solitons, peakons and cuspons), periodic ones (smooth, peakons and cuspons) and kinks. In the case of some special polynomials the solutions are expressed up to the inverse map by Gauss and Appell's hypergeometric functions, by some elliptic functions of Jacobi and Weierstrass types etc. In each case the above mentioned solutions of NLSE are obtained into integral form. [ABSTRACT FROM AUTHOR]

The energy for each level of the hydrogen atom was triggered by Neils Bohr, where the amount of energy decreases at every level of orbit. The value of energy depends on the principal quantum number which only applies to atoms with the s subshell of an atom with the same characteristics as the hydrogen atom. With the advancement of science, specifically in quantum physics, many potential equations aimed to solve the time-independent Schrodinger equation with the solution of the estimated value of energy. One of the applicable potential equations is the Hulthen potential which is explained about the potential at a small distance and only applies to atoms and micro-scale objects in physics and only has a solution at the azimuth quantum number value l = 0 or groups of atoms subshell s. Using the confluent hypergeometric function, the combination of the Hulthén potential and time-independent Schrödinger equation generates the estimated energy value of the hydrogen atom level, which can be used as a comparison with the estimated energy value of Bohr's hydrogen atom. This study found that the comparison between Hulthén and Bohr energy level lies in the same ratio 1:1 [ABSTRACT FROM AUTHOR]

The Bohr Mottelson model investigates the collective behavior of atomic nucleus [1]. The collective models are the combination of liquid drop model and shell model [2]. It is used to describes rotational and vibrational of the nucleus and also the deformed nucleus that corresponds to the excitation energy [3]. By solving the Bohr Mottelson, properties and mechanism involved in atomic nucleus can be obtained, such as energy spectrum and shape phase transitions [4]. The Bohr Mottelson has been solved for modified Davidson potential [5], Eckart potential [2], Kratzer potential [6], Killingbeck potential [7], Hulthen plus ring shape potential and three dimensional harmonic oscillator potential [8]. This study, the Bohr Mottelson equation is solved for modified Wood Saxon potential in spherical coordinates. By using hypergeometric method, the energy and the wave function of Bohr Mottelson equation were obtained. Numerically, the energy spectrum was calculated by applying energy equation in MATLAB R2013A software. While, the wave functions were investigated using hypergeometric method. The results show that the energy spectrum was increased by the increase of quantum numbers (n and L). The unnormalized wave functions were expressed in hypergeometric terms. [ABSTRACT FROM AUTHOR]

In the present article the author develop the solution of Fractional Kinetic Equation in a new and further generalized form by involving the ζ-Gauss Hypergeometric Functions as Kinetic Equations are having great importance in certain astrophysical problems. The change of chemical composition in star like the sun can be computed by this new generalized form of Kinetic equations. The main fold generality of the ζ-Gauss Hypergeometric Functions is discussed in terms of the solution of the Fractional Kinetic Equation. Special case involving the Gauss Hypergeometric function are also considered. The obtained results imply more precisely the known results and easily computable solution can also be established by the given results. [ABSTRACT FROM AUTHOR]

This paper is a continuation, as Part 2, of the recent author’s papers [18], [19], [20], where we have emphasized on our general and unified approach to evaluate operators of fractional calculus of a very general class of special functions. Namely, we have results for images of the Wright generalized hypergeometric functions (W. g.h.f.-s) under the operators of classical and generalized fractional calculus. Thus, great part of results published by other authors in numerous papers (for part of them - see references in the above mentioned 3 papers and also, herein) come as immediate particular cases. It is because the special functions considered there are all of them Wright g.h.f.-s, and the FC operators like the Riemann-Liouville (RL), Erdélyi-Kober (EK), Saigo, Marichev-Saigo-Maeda (MSM), etc., are all of them particular cases of the operators of Generalized Fractional Calculus (GFC), [11]. We gave previously long list of illustrative examples for the efficiency of the general approach, and now continue it. Recently, some authors repeated the job to evaluate FC operators of some special functions which they introduced and considered as “new” ones. Among them are some examples of the so-called k-analogues of the Bessel and Mittag-Leffler functions, generalized multi-index Bessel and Mittag-Leffler functions, K- and M-series and S -functions. In Section 5 we show that all these are just cases, again, of the Wright generalized hypergeometric function. Then, the results provided by the mentioned authors can come easily from our general ones. [ABSTRACT FROM AUTHOR]

The study of 300 years old history of quantum calculus or q−calculus or q−disease, since the Bernoulli and Euler, is often considered to be one of the most difficult subjects to engage in mathematics. Nowadays there is a rapid growth of activities in the area of q−calculus due to its applications in various fields such as mathematics, mechanics, and physics. The history of study of q−calculus may be illustrated by its wide variety of applications in quantum mechanics, analytic number theory, theta functions, hypergeometric functions, theory of finite differences, gamma function theory, Bernoulli and Euler polynomials, Mock theta functions, combinatorics, umbral calculus, multiple hypergeometric functions, Sobolev spaces, operator theory, and more recently in the theory of analytic and harmonic univalent functions. In q−calculus, we are generally interested in q−analogues that arise naturally, rather than in arbitrarily contriving q−analogues of known results. While focusing on excitement and romance with development of q−calculus and its applications in certain fields of mathematical sciences and physics, we will also look at q−analogues of some of the recent results in geometric function theory and, in particular, theory of analytic and harmonic univalent functions in the unit disc. [ABSTRACT FROM AUTHOR]

In the stationary regime, configurations of corrected Kompaneets equation have been analyzed within a theoretically approach and the spectrum evolution for the scattered photons has been derived. A particular attention has been granted to the impact of the nonlinearity on the spectrum distorsions, mainly to the closed form special solutions expressed in terms of Heun functions. In the nonrelativistic limit, stationary solutions of the Kompaneets equation bring into attention the Heun double confluent functions. On the other side, when various type of corrections come into play, the photon distribution functions are governed by the Heun triconfluent functions. In magnetar's magnetosphere, where the resonant Compton scattering is more important than the Compton one, the photon number density can be written in terms of Hermite and confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

A new exact solution of the overdetermined system of Oberbeck-Boussinesq equations is obtained. The shear flow of a non-uniformly rotating viscous incompressible fluid is considered. It is shown that the exact solutions of Oberbeck-Boussinesq equations are expressed by means of hypergeometric functions. [ABSTRACT FROM AUTHOR]

The Hypergeometric method was used to solve the Klein-Gordon equation with Trigonometric Poschl-Teller and Trigonometric Scarf II potentials. By using variable and parameter substitutions, Klein-Gordon equation for spin symmetry case was reduced into second-order differential equation. The radial part of Klein-Gordon equation with radial trigonometric Poschl-Teller potentials was solved in the scheme of centrifugal approximation by using hypergeometric method. The relativistic energy equation was obtained and calculated numerically by using Matlab R2013 software. On the other hand, wave function equation was obtained and visualized by Matlab R2013 software. The energy value depends on n, k, and μ values, while the quantum number and mass have effect for distribution of particle in visualization of wave function. [ABSTRACT FROM AUTHOR]

The Hypergeometric method was used to describe radial equations by probeneus approaximation at a singular regular point. Schrodinger equations radial part was reduced into differential equations with variable substitution. By using q-deformed momentum parameter and Coulomb potential as well as variable substitution, the second order of Schrodinger equation with centrifugal approach for q-deformed momentum in Coulomb potential can be solved with Hypergeometric method. The analytical result of energy spectrum and wave function with quantum number and qdeformed variations, while the numerical result for energy spectrum can be calculated by using Matlab 2013 software. [ABSTRACT FROM AUTHOR]

The method of special functions and heat polynomials for the solution of problems for the heat equation in domains with moving boundaries is developed for the generalized and axisymmetric heat equations. The corresponding generating and associated functions are introduced. Examples of application of this method for the solution of some problems are presented. [ABSTRACT FROM AUTHOR]

The present paper deals with multi-interval Sturm-Liouville equations with eigenparameter dependent boundary-transmission conditions. Such type of problems cannot be treated with the usual techniques within the standard framework os classical Sturmian theory. It is well-known that any eigenvalue of the classical Sturm-Liouville problems can be related to its eigenfunction by the Rayleigh quotient and some useful results can be obtained from the Rayleigh quotient without solving the differential equation. For instance, it can be quite useful in estimating the eigenvalues. In this study we present a new technique for investigation some computational aspects of the eigenvalues. Particularly, we give an operator-pencil formulation of the problem and establish lower bound estimation for eigenvalues by using modified Rayleigh quotient. [ABSTRACT FROM AUTHOR]

In this article, we will give theorems for the discrete fractional solutions of the homogeneous and nonhomogeneous associated Laguerre equation by using discrete fractional nabla operator. [ABSTRACT FROM AUTHOR]

One of the most popular research interests of science and engineering is the fractional calculus theory in recent times. Discrete fractional calculus has also an important position in fractional calculus. In this work, we acquire new discrete fractional solutions of the homogeneous and non homogeneous Legendre differential equation by using discrete fractional nabla operator. [ABSTRACT FROM AUTHOR]

In this paper we point out on some author's ideas and developments, combined with few basic classical results, that show how one can do the task posed in the title at once, and in rather general case: for both operators of generalized fractional calculus and generalized hypergeometric functions. In this way, the greater part of the results in the publications mentioned in References (and many others not in this limited list) are well predicted and come just as very special cases of the discussed general scheme. [ABSTRACT FROM AUTHOR]

Studying a special system of differential equations in the separate production of the second order is solved by the degenerate hypergeometric function reducing to the Bessel functions of two variables. To construct a solution of this system near regular and irregular singularities, we use the method of Frobenius-Latysheva applying the concepts of rank and antirank. There is proved the basic theorem that establishes the existence of four linearly independent solutions of studying system type of Bessel. To prove the existence of normal-regular solutions we establish necessary conditions for the existence of such solutions.The existence and convergence of a normally regular solution are shown using the notion of rank and antirank. [ABSTRACT FROM AUTHOR]

The possibilities of constructing normal-regular solutions of a system consisting of three partial differential equations of the second order are studied by the Frobenius-Latysheva method. The method of determining unknown coefficients is shown and the relationship of the studied system with the system, which solution is Laguerre's polynomial of three variables is indicated. The generalization of the Frobenius-Latysheva method to the case of a system consisting of three equations makes it possible to clarify the relationship of such systems, which solutions are special functions of three variables. These systems include the functions of Whittaker and Bessel, 205 special functions of three variables from the list of M.Srivastava and P.W. Carlsson, as well as orthogonal polynomials of three variables. All this contributes to the further development of the analytic theory of systems consisting of three partial differential equations of the second order. [ABSTRACT FROM AUTHOR]

In this article, we derive a new class of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution) involving some suitably normalized meromorphically q-Hypergeometric functions. A characterization property giving the coefficient bounds is obtained. The other related properties, which are studied in this paper, include distortion and the radii of meromorphic starlikeness and meromorphic convexity. [ABSTRACT FROM AUTHOR]

In this paper, we examine the hydromagnetic boundary layer flow and heat transfer characteristics of a laminar nanoliquid film over an unsteady stretching sheet is presented. The highly nonlinear partial differential equations governing flow and heat transport are simplified using similarity transformation. The analytical solutions of the resulting ODEs are obtained for some special case of nano liquid film using hypergeometric power series functions, and from which the analytical solutions of the original problem are presented. The influence of pertinent parameters such as the magnetic parameter, the solid volume fraction of nanoparticles and the type of nanofluid on the flow, heat transfer, Nusselt number and skin friction coefficient is discussed analytically. [ABSTRACT FROM AUTHOR]

We introduce a new subclass of meromorphically analytic functions, which is defined by means of a Hadamard product or convolution. A characterization property such as the coefficient bound is obtained for this class. The other related properties, which are investigated in this paper, include the distortion and the radius of starlikeness. We also consider several applications of our main results to the generalized hypergeometric functions. [ABSTRACT FROM AUTHOR]