Some limit formulas for the Gamma and Psi (or Digamma) functions at their singularities

Both the Gamma function Gamma( z) and the Psi (or Digamma) function Psi(z) are meromorphically continued to the whole complex z-plane with singularities (which are simple poles) at z = -k for non-positive integer values of k. Here, in this presentation, we derive the following (presumably new) limit formulas: lim z ->-k Gamma(nz)/Gamma(qz) = (-1)((n-q)k) (.)q/n . (qk)!/(nk)! and lim z ->-k Psi(nz)/Psi(qz) = q/n for positive integer values of n and q, k being a non-negative integer. One of the above limit formulas is used here to partition the singularities of Gamma(z) into the first set of singularities (at z = 0, -1, -2, -3) and the second set of singularities (at z = -4, -5, -6,...). Some other closely-related formulas are also considered.