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Abelian means

Suppose λ = { λ, λ, λ,... } is a strictly increasing sequence tending towards infinity, and that λ ≥ 0. Suppose

converges for all real numbers x > 0. Then the Abelian mean A is defined as

More generally, if the series for f only converges for large x but can be analytically continued to all positive real x, then one can still define the sum of the divergent series by the limit above.

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