Generalised heat coefficients and associated spectral zeta functions on complex projective spaces Pⁿ(ℂ)

ABSTRACT

The heat invariants or the Minakshisundaram-Pleijel heat coefficients $a_k^n=a_k^n\left(M\right)$ describe the residues of the spectral zeta functions on any compact Riemannian manifold M. In this paper, we first give an explicit description of the heat coefficients $a_k^n=a_k^n\left({\mathbf{P}}^n\right(\mathbb{C}\left)\right)$ via the Jacobi theta functions and their higher order derivatives, and then express these Minakshisundaram-Pleijel coefficients in terms of the residues of the associated zeta functions ${\zeta }_{{\mathbf{P}}^n\left(\mathbb{C}\right)}$. Different interesting formulae are obtained for these zeta functions. A new class of heat coefficients, the Maclaurin heat coefficients $b_{2m}^n=b_{2m}^n\left(t\right)$ ($t>0,m\ge 0$) (i.e. the coefficients appearing in the Maclaurin expansion of the heat kernel ${\mathsf{H}}_{{\mathbf{P}}^n\left(\mathbb{C}\right)}(t,\theta )$), are explicitly studied in terms of the classical and generalised Minakshisundaram-Pleijel coefficients $a_k^n$ and ${\mathsf{a}}_{k,j}^{n,m}$ ($1\le j\le m$) respectively. Remarkable asymptotic expansions for the Maclaurin spectral functions $b_{2m}^n\left(t\right)$ are established. We also introduce and construct new zeta functions ${\mathsf{Z}}_{{\mathbf{P}}^n\left(\mathbb{C}\right)}^m$ associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions are explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.

COMMUNICATED BY:

• D. Khavinson

KEYWORDS:

• Maclaurin heat coefficients
• generalised heat trace
• Minakshisundaram-Pleijel heat coefficients
• generalised Minakshisundaram-Pleijel heat coefficients
• spectral zeta function
• generalised Minakshisundaram-Pleijel zeta function

AMS SUBJECT CLASSIFICATIONS:

• 33C45
• 35A08
• 35C05
• 35C10
• 35C15

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Richard Olu Awonusika  http://orcid.org/0000-0001-9504-082X