Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 011, 32 pages      arXiv:1609.02525
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

Series Solutions of the Non-Stationary Heun Equation

Farrokh Atai ab and Edwin Langmann a
a) Department of Physics, KTH Royal Institute of Technology, SE-10691 Stockholm, Sweden
b) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received October 10, 2017, in final form February 08, 2018; Published online February 16, 2018

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.

Key words: Heun equation; Lamé equation; Kernel functions; quantum Painlevé VI; perturbation theory.

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