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 SIGMA, 2018, Volume 14, 011, 32 pages (Mi sigma1310)

Series Solutions of the Non-Stationary Heun Equation

Farrokh Ataiab, Edwin Langmannb

a Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
b Department of Physics, KTH Royal Institute of Technology, SE-10691 Stockholm, Sweden

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.

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

 Funding Agency Grant Number Stiftelsen Olle Engkvist Byggmästare 184-0573 We gratefully acknowledge partial financial support by the Stiftelse Olle Engkvist Byggmästare (contract 184-0573).

DOI: https://doi.org/10.3842/SIGMA.2018.011

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MSC: 33E20; 81Q05; 16R60
Received: October 10, 2017; in final form February 8, 2018; Published online February 16, 2018
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Citation: Farrokh Atai, Edwin Langmann, “Series Solutions of the Non-Stationary Heun Equation”, SIGMA, 14 (2018), 011, 32 pp.

Citation in format AMSBIB
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