Maclaurin Heat Coefficients and Associated Zeta Functions on Quaternionic Projective Spaces Pn(H) (n ≥ 1)

Abstract

Abstract The heat invariants or the Minakshisundaram-Pleijel heat coefficients a k n = a k n (M) (k ≥ 0) describe the asymptotic expansion of the heat kernel HM on any N = 4 n-dimensional (n ≥ 1) compact Riemannian manifold M; associated with the coefficients a k n is the Minakshisundaram-Pleijel zeta function ζM = ζM(s) (s ∈ C). In this paper, we introduce and study a new class of heat coefficients, namely, the Maclaurin heat coefficients b 2 m n = b 2 m n ( t ) (t > 0 , m 0) (i.e., the coefficients appearing in the Maclaurin expansion of the heat kernel HM(t, θ)) in terms of the classical and generalised Minakshisundaram-Pleijel coefficients a k n and a k , j n , m = a k , j n , m (M) (0 ⩽ j ⩽ m) respectively, when M = P n (H) (n ≥ 1), a quaternionic projective space. Remarkable asymptotic expansions for the Maclaurin spectral functions b 2 m n ( t ) are established. We also introduce and construct new zeta functions Z p n ( H ) m (m ≥ 0) associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions can be explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.