On Jacobi polynomials (P (a,ß) k : a, ß > -1) and Maclaurin spectral functions on rank one symmetric spaces

The Maclaurin spectral functions associated with the development of the heat kernel on compact rank one symmetric spaces are analysed. Relations with various invariants most notably the heat trace, the Minakshisundaram–Pleijel heat coefficients and the spectral residues are carefully examined and a precise formulation as well as asymptotics (t & 0) in terms of the celebrated Jacobi theta functions is represented. A natural class of polynomials and power series encoding structural properties of the heat kernel are introduced and further studied.