This is a reference request for a theorem I thought I had read in a book by Steven Krantz, but I can no longer find it.
Searching for Mittag-Leffler star, I can find references to the following result. Suppose we have a complex analytic function $$ f(z) = \sum_{n=0}^\infty a_n (z-w)^n $$ with some positive radius of convergence in a ball $B(w,r) \subseteq \mathbb{C}$.
We say $D\subseteq \mathbb{C}$ is starlike around $w$ if $w\in D$ and also for every $w' \in D$ we have the line connecting $w$ to $w'$ in $\mathbb{C}$ is contained in $D$. We say that $f$ can be analytically continued to a starlike domain $D$ if there exists some analytic $f_D:D \to \mathbb{C}$ such that $f_D|_{B(w,r)} = f$.
Then there exists a sequence $k_n$ and coefficients $c_0^{(n)},\ldots,c_{k_n}^{(n)}$, independent of $f$, such that $$ f(z) = \sum_{n=0}^\infty \sum_{m=0}^{k_n} c_m^{(n)} a_m (z-w)^m $$ and where this sum converges and agrees with the analytic continuation of $f$ on any starlike domain around $w$.
My recollection is that this analytic continuation can also be accomplished using something like Mittag-Leffler summation. As I remember, we can define $$ f_\varepsilon(z) = \sum_{n=0}^\infty \frac{a_n}{\Gamma(\varepsilon n + 1)} (z-w)^n $$ which is entire, and then $f_\epsilon(z) \xrightarrow {\varepsilon\to0} f_D(z)$ on any starlike $D$. But I cannot find a proof or name of this last statement.
