  We study the analogue of polynomials (solutions to $\Delta^{n+1} u =0$ for
some $n$) on the Sierpinski gasket ($SG$) with respect to a family of
symmetric, self-similar Laplacians constructed by Fang, King, Lee, and
Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on
the polynomials with respect to the standard Kigami Laplacian. We define a
basis for the space of polynomials, the monomials, characterized by the
property that a certain "derivative" is 1 at one of the boundary points, while
all other "derivatives" vanish, and we compute the values of the monomials at
the boundary points of $SG$. We then present some data which suggest surprising
relationships between the values of the monomials at the boundary and certain
Neumann eigenvalues of the family of symmetric self-similar Laplacians.
Surprisingly, the results for the general case are quite different from the
results for the Kigami Laplacian.
