Universal Taylor series with respect to a prescribed subsequence

For a holomorphic function $f$ in the open unit disc $\mathbb{D}$ and $\zeta \in \mathbb{D}$, $S_n(f,\zeta )$ denotes the $n$-th partial sum of the Taylor development of $f$ at $\zeta$. Given an increasing sequence of positive integers $\mu =({\mu }_n)$, we consider the classes $\mathcal{U}(\mathbb{D},\zeta )$ (resp. ${\mathcal{U}}^{(\mu )}(\mathbb{D},\zeta )$) of such functions $f$ such that the partial sums $\{S_n(f,\zeta ):n=1,2,\dots \}$ (resp. $\{S_{{\mu }_n}(f,\zeta ):n=1,2,\dots \}$) approximate all polynomials uniformly on the compact sets $K\subset \{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\ge 1\}$ with connected complement. We show that these two classes of universal Taylor series coincide if and only if ${\mathrm{lim\hspace{0.17em}sup}}_n\left(\frac{{\mu }_{n+1}}{{\mu }_n}\right)<+\mathrm{\infty }$. In the same spirit, we prove that, for $\zeta \mathrm{\ne }0,$ we have the equality ${\mathcal{U}}^{(\mu )}(\mathbb{D},\zeta )={\mathcal{U}}^{(\mu )}(\mathbb{D},0)$ if and only if ${\mathrm{lim\hspace{0.17em}sup}}_n\left(\frac{{\mu }_{n+1}}{{\mu }_n}\right)<+\mathrm{\infty }$. Finally we deal with the case of real universal Taylor series.