Remarks on Lubbock's summation formulae

TL;DR

This paper explores the coefficients in Lubbock's summation formulae, linking them to generalized Bernoulli polynomials and applying them to problems in quantum field theory and trigonometric summations.

Contribution

It generalizes Lubbock's summation formulae using Bernoulli polynomials and introduces a formal Lubbock formula for delta operators, connecting to various mathematical and physical applications.

Findings

Coefficients are generalized Bernoulli polynomials.

Application to subdivision problems in quantum field theory.

Introduction of a formal Lubbock formula for delta operators.

Abstract

The coefficients occurring in summation formulae of the Lubbock type are shown to be generalised Bernoulli polynomials which turn up in subdivision questions such as quantum field theory around a conical singularity and on spherical lunes. An image interpretation is made, generating functions are brought in and some trigonometric summations encountered. A formal Lubbock formula is introduced for a general delta operator.