  We study the bi-graded Hilbert function of ideals of general fat points with
same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the
multiprojective-affine-projective method introduced by the second author in
previous works with A.V. Geramita and A. Gimigliano where they solved the case
of double points. In this way, we compute the Hilbert function when the
smallest entry of the bi-degree is at most the multiplicity of the points. Our
second tool is the differential Horace method introduced by J. Alexander and A.
Hirschowitz to study the Hilbert function of sets of fat points in standard
projective spaces. In this way, we compute the entire bi-graded Hilbert
function in the case of triple points.
