Solve the asymmetric friendly-walk functional equations

Solve the functional equations for three friendly directed walks with asymmetric pairwise contact interactions, whose symmetry breaking prevents the determination of a sufficient system of equations for the generating function.

Background

The paper contrasts the solvable symmetric model of three friendly directed walks in a watermelon configuration with the asymmetric case, in which the interaction between different pairs of walks may differ. The authors report that their preliminary attempts could not solve the corresponding functional equations because breaking the symmetry appeared to leave too few equations, and because the generating function in the symmetric case was D-finite and therefore did not provide a straightforward guide to the asymmetric solution.