Mittag-Leffler summation
In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Gösta Mittag-Leffler (1908) == Definition == Let y ( z ) = ∑ k = 0 ∞ y k z k {\displaystyle y(z)=\sum _{k=0}^{\infty }y_{k}z^{k}} be a formal power series in z. Define the transform B α y {\displaystyle {\mathcal {B}}_{\alpha }y} of y {\displaystyle y} by B α y ( t ) ≡ ∑ k = 0 ∞ y k Γ ( 1 + α k ) t k {\displaystyle {\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 α → 0 B α y ( z ) {\displaystyle \lim _{\alpha \rightarrow 0}{\mathcal {B}}_{\alpha }y(z)} if each sum converges and the limit exists.