Series Solutions of the Non-Stationary Heun Equation

Abstract

Abstract. We consider the non-stationary Heun equation, also known as quantum Painlevé VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the problem to solve this equation into a differential-difference equation which, as we show, can be solved by efficient recursive algorithms. We thus obtain series representations of solutions which provide elliptic generalizations of the Jacobi polynomials. These series reproduce, in a limiting case, a perturbative solution of the Heun equation due to Takemura, but our method is different in that we expand in non-conventional basis functions that allow us to obtain explicit formulas to all orders; in particular, for special parameter values, our series reduce to a single term.

Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation

“…The special case n = 2 of our results is equivalent to an integral representation of elliptic generalizations of the Gegenbauer polynomials obtained in [8]; see [9,Section 3.2] in the special case whereg ν = g for all ν = 0, 1, 2, 3 and [10] for similar such results. The key to our generalization of this result are generalized kernel function identities for the eCS model obtained in [11].…”

“…, λ m , 0 n−m ) ∈ P n , i.e., we identify partitions that differ only by a string of zeros. 9 Note that λ − (λ n n ) always is a partition under the stated conditions.…”

“…As explained in more detail below, results in [7,9,23,24,25,26] suggest to us that, for arbitrary n ∈ Z ≥1 , g ∈ C, κ ∈ C \ {0} and λ ∈ C n , there exist functions P These functions are analytic functions of (z, p) and (λ, κ) in suitable regions ⊂ C n+1 and, for partitions λ, they are symmetric functions in z that reduce to the Jack polynomials P (1/g) λ,n (z) in the limit p → 0. We believe that there is a beautiful mathematical theory of this two-parameter deformation of the Jack polynomials which, at this point, is only partly developed.…”

“…(ii) We constructed series representations of such functions P (1/g) λ;κ,n (z; p) for the simplest non-trivial case n = 2 in [9]. 11 (iii) A construction of singular series solutions (in the sense of [25]) of the non-stationary eCS equation for κ = 0 and arbitrary n ∈ Z ≥2 by one of us in [26], which is straightforward to extend this construction to κ = 0.…”

“…11 (iii) A construction of singular series solutions (in the sense of [25]) of the non-stationary eCS equation for κ = 0 and arbitrary n ∈ Z ≥2 by one of us in [26], which is straightforward to extend this construction to κ = 0. In fact, this construction is easier for non-zero κ [9]: first, if κ = 0, one need not compute the eigenvalue E(τ ) in (18) and, second, if ℑ(κ) = 0, it is easy to prove absolute convergence of these singular series solutions in a suitable domain ⊂ C n+1 using the method in [26]. Moreover, using the kernel functions in (38), one can transform these singular solutions to such described in the previous paragraph (this is explained in [25]), but the latter is restricted to κ = kg with k ∈ Z and, for k > 0, these solutions are incomplete (for k < 0 they are over-complete).…”