Separation of Variables ( Conduction)

I am working on this separation of the variable problem, and given that this square conducting pipe of side length 2a, while also parallel to the z-axis. They give that:

$$
V(x, y) = sum_{k=0}^{infty}[(A_ksinh(kx)+B_kcosh(kx))(C_ksin(ky)+D_kcos(ky)) + (E_ksinh(ky)+F_kcosh(ky))(G_ksin(kx)+H_kcos(kx)) ]
$$

The boundary conditions for the electric potential I have found for $V(x, y)$ are $V (−a, y) =0, V (a, y) =0,V (x,−a) = V (x, a) =V_kcos(pi x/a)$.
Trying to use boundary conditions, I need help understanding what happens to the coefficients like which ones go to zero for $x=pm a$ and when $y=pm a$ boundary conditions. Overall how does $k$ change with these boundaries conditions?