Mittag Leffler summation - Alchetron, the free social encyclopedia

In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Mittag-Leffler (1908)

Definition

Let

y ( z ) = ∑ k = 0 ∞ y k z k

be a formal power series in z.

Define the transform B α y of y by

B α y ( t ) ≡ ∑ k = 0 ∞ y k Γ ( 1 + α k ) t k

Then the Mittag-Leffler sum of y is given by

lim α → 0 B α y ( z )

if each sum converges and the limit exists.

A closely related summation method, also called Mittag-Leffler summation, is given as follows (Sansone & Gerretsen 1960). Suppose that the Borel transform converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Mittag-Leffler sum of y is given by

∫ 0 ∞ e − t B α y ( t α z ) d t

When α = 1 this is the same as Borel summation.

References

Mittag-Leffler summation Wikipedia

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