Related papers: Quantum Field Theory Applications of Heun Type Fun…
The usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann…
In this paper, we study rational functions of $q$-Racah type and a multivariate extension, using representation theory of $\mathcal U_q(\mathfrak{sl}_2)$. Eigenfunctions of twisted primitive elements in $\mathcal U_q(\mathfrak{su}_2)$ can…
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…
This paper addresses a general method of polynomial transformation of hypergeometric equations. Examples of some classical special equations of mathematical physics are generated. Heun's equation and exceptional Jacobi polynomials are also…
A formula for evaluating the quadratic normalization integrals of orthogonal Heun functions over the real interval 0 <= x <= 1 is derived using a simple limiting procedure based upon the associated differential equation. The resulting…
We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of differential operators of order two. They depend on complex parameters $\lambda$, $\mu$, $n$. Its…
We develop a non-relativistic quantum field theory of electrons and nuclei based on the Coulomb Hamiltonian. We derive the exact equations of motion and write these equations in the form of Hedin's equations for all species of identical…
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in…
Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions.…
We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(\lambda)=\log(\sigma_k(\lambda))$ on a subset of the positive cone $\Gamma_n=\{\lambda=(\lambda_{1}, \lambda_{2}, \cdots,\lambda_{n})\in \mathbb{R}^n;…
We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform…
In this article we show for the first time the role played by the hypergeneralized Heun equation (HHE) in the context of Quantum Field Theory in curved space-times. More precisely, we find suitable transformations relating the separated…
For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess…
It is shown that Schr\"odinger equation with combination of three potentials U = - {\alpha} r^{-1} + {\beta} r + kr^{2}, Coulomb, linear and harmonic, the potential often used to describe quarkonium, is reduced to a bi-confluent Heun…
A few 2+1-dimensional equations belonging to the KP and modified KP hierarchies are shown to be sufficient to provide a unified picture of all the integrable cases of the cubic and quartic H\'enon-Heiles Hamiltonians.
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…
We develop a method to determine the eigenvalues and eigenfunctions of two-boson Hamiltonians include a wide class of quantum optical models. The quantum Hamiltonians have been transformed in the form of the one variable differential…
We derive out a complete series expression of Hamiltonian eigenvalues without any approximation and cut in the general quantum systems based on Wang's formal framework \cite{wang1}. In particular, we then propose a calculating approach of…
We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that…
In the present paper, we work out the eigenfunctions of spinless particles bound in a one-dimensional linear finite range, attractive potential well, treating it as a time-like component of a four-vector. We show that the one-dimensional…