Universal partial sums of Taylor series as functions of the centre of expansion

Abstract

V. Nestoridis conjectured that if $\Omega$ is a simply connected subset of $\mathbb{C}$ that does not contain $0$ and $S(\Omega)$ is the set of all functions $f\in \mathcal{H}(\Omega)$ with the property that the set $\left\{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right\}$ is dense in $\mathcal{H}(\Omega)$, then $S(\Omega)$ is a dense $G_\delta$ set in $\mathcal{H}(\Omega)$. We answer the conjecture in the affirmative in the special case where $\Omega$ is an open disc $D(z_0,r)$ that does not contain $0$.