Orthogonal Polynomials on Bubble-Diamond Fractals

We develop a theory of polynomials and, in particular, an analog of the theory of Legendre orthogonal polynomials on the bubble-diamond fractals, a class of fractal sets that can be viewed as the completion of a limit of a sequence of finite graph approximations. In this setting, a polynomial of degree j can be viewed as a multiharmonic function, a solution of the equation Δj+1u=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^{j+1}u=0$$\end{document}. We prove that the sequence of orthogonal polynomials we construct obeys a three-term recursion formula.