Solve the asymmetric watermelon and general catalytic-variable cases

Solve the functional equation for the asymmetric three-osculating-walk model in the watermelon specialization r=s=0 and, more generally, for arbitrary catalytic variables r and s using a kernel-method approach or another exact method.

Background

The paper obtains an algebraic equation for the generating function of the asymmetric star model at r=s=1, but does not solve the corresponding functional equation for the watermelon case or for general endpoint-distance variables. The unresolved cases would extend the exact treatment beyond the star generating function and clarify whether the broken r↔s symmetry is the fundamental obstruction to applying the kernel method.

References

We have not been able to use the kernel method to directly solve the above equation~3ow_main_fn_eqn for the watermelon case, namely r=s=0, nor for general r,s by using any variant of the kernel method.

— Exact solution of asymmetric gelation between three walks on the square lattice  (2507.17111 - Owczarek et al., 23 Jul 2025) in Section 5, subsection “Three walks: asymmetric interactions for stars r=s=1”