Related papers: New relations for the derivative of the confluent …
In this paper, we give the matrix version of Horn's hypergeometric function and its confluent cases. We also discuss the regions of convergence, the system of matrix differential equations of bilateral type, differential formulae and…
The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the…
We present in total fifteen potentials for which the stationary Klein-Gordon equation is solvable in terms of the confluent Heun functions. Because of the symmetry of the confluent Heun equation with respect to the transposition of its…
In the present article we introduce and study a novel type of solutions of the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Felix Klein symmetric form…
We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlev\'{e} IV that admits special solutions. The BHC degenerates when either the Stokes' data for the…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function…
We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. While the Dirac equation written in the background of Nutku helicoid metric yields Mathieu functions as its solutions in…
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…
The reducible double confluent Heun equation (DCHE) is the only DCHE whose general symmetric unfolding leads to a Fuchsian equation. Contrary to general Heun equation the unfolded Fuchsian equation has 5 singular points :…
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 functional relation of the Hurwitz zeta function is proved by using the connection problem of the confluent hypergeometric equation.
A new bi-parametric $su(1,1)$ algebraization of the Heun class of equations is explored. This yields additional quasi-polynomial solutions of the form $\{z^{\alpha}P_N(z): \ \alpha \in \mathbb{C}, \ N \in \mathbb{N}_0\}$ to the General Heun…
Humbert confluent hypergeometric functions of two variables arise in many problems of mathematical physics and applied analysis, yet their behavior with respect to parameters has not been systematically studied. In this paper we investigate…
We present a numerical implementation of the recently developed unconditionally convergent representation of general Heun functions as integral series. We produce two codes in Python available for download, one of which is especially aimed…
We study a novel type of solutions of the general Heun's equation, based on its symmetric form. We derive the symmetry group of this equation which is a proper extension of the Mobius group. The new series solution treat simultaneously and…
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…
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 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…