Related papers: Polynomial solutions of $q$-Heun equation and ultr…
It is known that the q-Heun equation has polynomial-type solutions in some special cases, and the condition for the accessory parameter E is described by the roots of the spectral polynomial. We investigate the spectral polynomial by…
The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding…
The well-known Heun equation has the form: Q(z)S''(z)+P(z)S'(z)+V(z)S(z)=0 where Q(z) is a cubic complex polynomial, P(z) and V(z) are polynomials of degrees at most 2 and 1 resp. One of the classical problems about the Heun equation is for…
The classical Heun equation has the form {Q(z) d^2/dz^2 +P(z) d/dz +V(z)}S(z)=0 where Q(z) is a cubic, P(z) at most quadratic and V(z) linear polynomials resp. In the second half of the 19-th century E.Heine and T.STieltjes initiated the…
We obtain special solutions of the $q$-Heun equation which are expressed as finite summations of $q$-hypergeometric functions. These solutions are obtained by considering the $q$-integral transformations of the polynomial-type solutions.
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{\mathbb R})$. As a consequence it is…
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…
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before…
We consider two quasi-exactly solvable problems in one dimension for which the Schr\"odinger equation can be converted to Heun's equation. We show that in neither case the Bender-Dunne polynomials form an orthogonal set. Using the…
We consider in this paper the Hahn polynomials and their application in numerical methods. The Hahn polynomials are classical discrete orthogonal polynomials. We analyse the behaviour of these polynomials in the context of spectral…
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…
We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…
We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the…
This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials $\{ P_n\}$. The quantum-mechanical wave function is the generating function for the $P_n (E)$,…
The present paper analyze the constraints on the confluent Heun type-equation, $(a_{3,1}r^2+a_{3,2}r)y"+(a_{2,0}r^2+a_{2,1}r+a_{2,2})y'-(\tau_{1,0}r+\tau_{1,1})y=0,$ where $|a_{3,1}|^2+|a_{3,2}|^2\neq 0, $ and $a_{i,j},i=3,2,1, j=0,1,2$ are…
The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by…
We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the…
A second-order differential (q-difference) eigenvalue equation is constructed whose solutions are generating functions of the dual (q-)Hahn polynomials. The fact is noticed that these generating functions are reduced to the (little…
After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the…
Bethe ansatz formulation is presented for several explicit examples of quasi exactly solvable difference equations of one degree of freedom which are introduced recently by one of the present authors. These equations are deformation of the…