Summation formula over the zeros of the associated Legendre function with a physical application

TL;DR

This paper derives a summation formula for the zeros of associated Legendre functions using the generalized Abel-Plana formula, with applications to quantum field theory in curved space, simplifying the calculation of the Wightman function.

Contribution

It introduces a new summation formula for zeros of associated Legendre functions and applies it to express the Wightman function in curved space with boundaries.

Findings

Derived a summation formula for zeros of associated Legendre functions.

Applied the formula to simplify the Wightman function in curved space.

Facilitated vacuum expectation value renormalization by separating boundary effects.

Abstract

Associated Legendre functions arise in many problems of mathematical physics. By using the generalized Abel-Plana formula, in this paper we derive a summation formula for the series over the zeros of the associated Legendre function of the first kind with respect to the degree. The summation formula for the series over the zeros of the Bessel function, previously discussed in the literature, is obtained as a limiting case. The Wightman function for a scalar field with general curvature coupling parameter is considered inside a spherical boundary on background of constant negative curvature space. The corresponding mode sum contains series over the zeros of the associated Legendre function. The application of the summation formula allows us to present the Wightman function in the form of the sum of two integrals. The first one corresponds to the Wightman function for the bulk geometry without boundaries and the second one is induced by the presence of the spherical shell. For points away from the boundary the latter is finite in the coincidence limit. In this way the renormalization of the vacuum expectation value of the field squared is reduced to that for the boundary-free part.