Related papers: Generalized confluent hypergeometric solutions of …
The first order nonlinear ODE \dot \phi(t) + \sin\phi(t)=q(t),q(t)=B+A\cos\omega t, where A,B,\omega are real constants, is considered, the transformation converting it to a second order linear homogeneous ODE with polynoimial coefficients…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
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…
Sch\"afke and Schmidt established that the asymptotics of the coefficients of the local solution to some linear differential equation is related to global structures of solutions. The Heun class equations have the accessory parameters, and…
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 review properties of confluent functions and the closely related Laguerre polynomials, and determine their bilinear integrals. As is well-known, these integrals are convergent only for a limited range of parameters. However, when one…
An analysis of the generalized confluent Heun equation $(\alpha_2r^2+\alpha_1r)\,y''+(\beta_2r^2+\beta_1r+\beta_0)\,y'-(\varepsilon_1r+\varepsilon_0)\,y=0$ in $d$-dimensional space, where $\{\alpha_i, \beta_i, \varepsilon_i\}$ are real…
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…
This paper explores the calculus of dual-valued functions and investigates the gamma function, beta function and generalized hypergeometric functions by incorporating dual numbers as parameters and variables. We examine its fundamental…
We review recent results on analytical properties (monotonicity and bounds) for ratios of contiguous functions of hypergeometric type. The cases of parabolic cylinder functions and modified Bessel functions have been discussed with…
The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index $n$ runs from minus to plus infinity [E. W. Leaver, J. Math.…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
By exploiting a recently developed connection between Heun's differential equation and the generalized associated Lam\'e equation, we not only recover the well known periodic solutions, but also obtain a large class of new, quasi-periodic…
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…
We give closed forms for several families of Heun functions related to classical entropies. By comparing two expressions of the same Heun function, we get several combinatorial identities generalizing some classical ones.
A certain special function of the generalized hypergeometric variety is shown to fulfill a host of useful noncommutative identities.
In this work, generalized hypergeometric functions for bicomplex argument is introduced and its convergence criteria is derived. Furthermore, integral representation of this function has been established. Moreover, quadratic transformation,…
We describe the close connection between the linear system for the sixth Painlev\'e equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the…
We obtain isomonodromic transformations for Heun's equation by generalizing Darboux transformation, and we find pairs and triplets of Heun's equation which have the same monodromy structure. By composing generalized Darboux transformations,…
The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one…