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
Sansone, Giovanni; Gerretsen, Johan (1960), Lectures on the theory of functions of a complex variable. I. Holomorphic functions, P. Noordhoff, Groningen,
MR0113988