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

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 H_M 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 ζ_M = ζ_M(s) (s ∈ C). In this paper, we introduce and study a new class of heat coefficients, namely, the Maclaurin heat coefficients $b_{2m}^n=b_{2m}^n\left(t\right)$ (t > 0 , m 0) (i.e., the coefficients appearing in the Maclaurin expansion of the heat kernel H_M(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 $b_{2m}^n\left(t\right)$ are established. We also introduce and construct new zeta functions $Z_{pn\left(H\right)}^m$ (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. (paper)