Divergent Series - Abelian Means

Abelian Means

Suppose λ = {λ₀, λ₁, λ₂, ...} is a strictly increasing sequence tending towards infinity, and that λ₀ ≥ 0. Recall that a_n = λ_n+1 − λ_n is the associated series whose partial sums form the sequence λ. Suppose

converges for all positive real numbers x. Then the Abelian mean A_λ is defined as

A series of this type is known as a generalized Dirichlet series; in applications to physics, this is known as the method of heat-kernel regularization.

Abelian means are regular, linear, and stable, but not always consistent between different choices of λ. However, some special cases are very important summation methods.