Belyi functions for hyperbolic hypergeometric-to-Heun transformations

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k1/k, 1/ℓ1/ℓ, 1/m1/m that satisfy k,ℓ,m∈Nk,ℓ,m∈N and the hyperbolic condition 1/k+1/ℓ+1/m<11/k+1/ℓ+1/m<1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.