Padé tables of a class of entire functions

Abstract:

It is shown that if the Maclaurin series coefficients $\left \{ {{a_j}} \right \}$ of an entire function satisfy a certain explicit condition, then there exists a sequence $\mathcal {S}$ of integers such that $[L/{M_L}] \to f$ locally uniformly in ${\mathbf {C}}$ as $L \to \infty ,L \in \mathcal {S}$, for all nonnegative integer sequences $\left \{ {{M_L}} \right \}_{L = 1}^\infty$. In particular, this condition is satisfied if the $\left \{ {{a_j}} \right \}$ approach 0 fast enough, or if a subsequence of the $\left \{ {{a_j}} \right \}$ behaves irregularly in a certain sense. Further, the functions satisfying this condition are dense in the space of entire functions with the topology of locally uniform convergence. Consequently, the set of entire functions satisfying the Baker-Gammel-Wills Conjecture is of the second category.