Let :y(z) = \sum_{k = 0}^\infty y_kz^k be a
formal power series in
z. Define the transform \mathcal{B}_\alpha y of y by :\mathcal{B}_\alpha y(t) \equiv \sum_{k=0}^\infty \frac{y_k}{\Gamma(1+\alpha k)}t^k Then the
Mittag-Leffler sum of
y is given by :\lim_{\alpha\rightarrow 0}\mathcal{B}_\alpha y( z) if each sum converges and the limit exists. A closely related summation method, also called Mittag-Leffler summation, is given as follows . Suppose that the Borel transform \mathcal{B}_1 y(z) 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 :\int_0^\infty e^{-t} \mathcal{B}_\alpha y(t^\alpha z) \, dt When
α = 1 this is the same as
Borel summation. ==See also==