Mittag-Leffler summation

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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[edit]

Let

be a formal power series in z.

Define the transform of by

Then the Mittag-Leffler sum of y is given by

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[when defined as?] 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

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

See also[edit]

References[edit]