Related papers: Generalized-hypergeometric solutions of the biconf…
A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical…
In this paper, we focus on clarifying the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions and providing another proof of the non-existence of algebraic solutions for equations…
Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful…
We obtain several degenerations of the $q$-Heun equation by considering the linear $q$-difference equations associated to several $q$-Painlev\'e equations. We establish definitions of the confluent $q$-Heun equation, the biconfluent…
A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic…
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 solve connection problem between fundamental solutions at singular points $0$ and $1$ for the generalized hypergeometric function, using analytic continuation of the integral representation. All connection coefficients are products of…
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…
In this paper, we aim to obtain a representation of Humbert's hy- pergeometric function in a series of Gauss's function 2F1. A few interesting results have also been deduced as special case of our main findings.
We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the…
In this paper, a quantum dot mathematical model based on a two-dimensional Schr\"odinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a…
We obtain some new formulae to compute the first derivative of confluent and biconfluent Heun functions under the minimal assumption of fixing only one parameter. These results together with the Lagrangian formulation of a general…
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.…
Firstly, we construct kernels of integral relations among solutions of the confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both cases we generate additional kernels by systematically applying substitutions of…
Hypergeometric solutions to the q-Painlev\'e equations are constructed by direct linearization of disrcrete Riccati equations. The decoupling factors are explicitly determined so that the linear systems give rise to q-hypergeometric…
Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.
We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple…
We discuss the level-crossing field configurations for which the quantum time-dependent two-state problem is solvable in terms of the confluent Heun functions. We show that these configurations belong to fifteen four-parametric families of…
Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by a Nevanlinna theory…
Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate…