Symmetric functions and the Vandermonde matrix

This work deduces the lower and the upper triangular factors of the inverse of the Vandermonde matrix using symmetric functions and combinatorial identities. The L and U matrices are in turn factored as bidiagonal matrices. The elements of the upper triangular matrices in both the Vandermonde matrix and its inverse are obtained recursively. The particular value x i =1+ q+⋯+ q i−1 in the indeterminates of the Vandermonde matrix is investigated and it leads to q-binomial and q-Stirling matrices. It is also shown that q-Stirling matrices may be obtained from the Pascal matrix.