Series Solutions of the Non-Stationary Heun Equation

TL;DR

This paper develops a recursive algorithm to solve the non-stationary Heun equation, producing series solutions that generalize Jacobi polynomials and include explicit formulas for all orders.

Contribution

It introduces a novel recursive method to obtain series solutions of the non-stationary Heun equation using a differential-difference approach, enabling explicit formulas and elliptic generalizations.

Findings

Series solutions generalize Jacobi polynomials.

Explicit formulas obtained for all orders.

Method reduces to a single term for special parameters.

Abstract

We consider the non-stationary Heun equation, also known as quantum Painlev\'e 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.