Related papers: Generalized hypergeometric solutions of the Heun e…
Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the…
In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some…
The Heun's equation is the Fuchsian equation of second order with four regular singularities. Heun functions generalize well-known special functions such as Spheroidal Wave, Lam\'{e}, Mathieu, hypergeometric-type functions, etc. The…
We consider the three most important equations of hypergeometric type, ${}_2F_1$, ${}_1F_1$ and ${}_1F_0$, in the so-called degenerate case. In this case one of the parameters, usually denoted $c$, is an integer and the standard basis of…
We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iteratated solutions to…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
The authors establish the necessary and sufficient conditions under which certain combinations of Gaussian hypergeometric function and elementary function are monotone in the parameter, which generalize the recent results of generalized…
Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…
We give the hypergeometric solutions of some algebraic equations including the general fifth degree equation.
The Heun's equation with its four regular singularities emerges in many applications in science. Despite the growing interest of the scientific community, the literature has many gaps in conceptual mathematical aspects of this equation.…
Recently found all the fundamental solutions of a multidimensional singular elliptic equation are expressed in terms of the well-known Lauricella hypergeometric function in many variables. In this paper, we find a unique solution of the…
In the paper we consider the Heun functions, which are solutions of the equation introduced by Karl Heun in 1889. The Heun functions generalize many known special functions and appear in many fields of modern physics. Evaluation of the…
Just as with the Gauss hypergeometric function, particular cases of the local Heun function can be Liouvillian (that is, "elementary") functions. One way to obtain these functions is by pull-back transformations of Gauss hypergeometric…
We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each…
A recently proposed algorithm to obtain global solutions of the double confluent Heun equation is applied to solve the quantum mechanical problem of finding the energies and wave functions of a particle bound in a potential sum of a…
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…
In the paper we deal with the Heun functions --- solutions of the Heun equation, which is the most general Fuchsian equation of second order with four regular singular points. Despite the increasing interest to the equation and numerous…
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy…