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Dedicated to S. L. LeeIn a recent generalization of the Bernstein polynomials, the approximated function / is evaluated at points spaced at intervals which are in geometric progression on [0,1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial B,f by a one-parameter family of polynomials Blf, where 0 < q < 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning Blf when / is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if / is increasing then &J is increasing, and if/ is convex then Blf is convex, generalizing well known results when q = 1. It is also shown that if/ is convex then, for any positive integer n, BJ < Blf for 0 < q < r < 1. This supplements the well known classical result that/ < B n / when/ is convex.1991 Mathematics subject classification: 41A10.

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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 valuex 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.