A formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomials

Abstract

The Jacobi polynomials $(\mathscr{P}^{(\alpha, \beta)}_k: k \ge 0, \alpha, \beta >-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the spherical or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.