  Some applications of a result, which is proved recently, is considered. We
first prove three determinantal identities concerning the binomial coefficient
and Stirling numbers of the first and the second kind. We also easily obtain
the inverse of the Vandermonde matrix. Then we derive a recurrence formula for
sums of powers, which is similar to the well-known Newton identity. In the last
section, we consider some sequences given by a homologous linear recurrence
equation.
  A determinantal identity for the Fibonacci numbers of higher order is proved.
  We finish with an expression of the generalized Vandermonde determinant in
terms of the standard Vandermonde determinant and elementary symmetric
polynomials.
