g[x_, i_, j_, p_, q_] = (x + p)^i*(x + q)^j;

f[u_, i_, j_, p_, q_] = SeriesCoefficient[g[x, i, j, p, q], {x, 0, u},
                            Assumptions -> u >= 0]

(*  p^i q^(j - u) Binomial[j, u] Hypergeometric2F1[-i, -u, 1 + j - u, q/p]  *)

Table[g[x, i, j, p, q] ==
    Sum[x^v*Limit[f[u, i, j, p, q], u -> v], {v, 0, i + j}],
         {i, 0, 3}, {j, 0, 7}] // Simplify

(*   {{True, True, True, True, True, True, True, True},
      {True, True, True, True, True, True, True, True},
      {True, True, True, True, True, True, True, True},
      {True, True, True, True, True, True, True, True}}   *)