# Polynomials on the Sierpinski Gasket with Respect to Different Laplacians which are Symmetric and Self-Similar

## TL;DR

This paper explores polynomial solutions related to various symmetric, self-similar Laplacians on the Sierpinski gasket, revealing unexpected differences from the classical Kigami Laplacian case.

## Contribution

It introduces a basis of polynomials for these Laplacians, characterizes them by boundary derivatives, and uncovers surprising boundary value relationships and eigenvalue connections.

## Key findings

- Defined monomials with specific boundary derivative properties
- Computed monomial boundary values for different Laplacians
- Discovered unexpected relationships between monomials and Neumann eigenvalues

## Abstract

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.