= E-function
{c}
{title2=$f(z)=\sum_{n\geq0}a_nz^n/n!$}
In the modern strict sense an E-function satisfies a nonzero linear differential equation over $\overline{\mathbb Q}(z)$, has algebraic coefficients $a_n$ with all conjugates bounded by $C^{n+1}$, and has common positive denominators $d_n\leq C^{n+1}$ making $d_na_0,\ldots,d_na_n$ algebraic integers. Siegel's original arithmetic growth condition permits, for each $\epsilon>0$, bounds of size $O_\epsilon(n^{\epsilon n})$ instead. The strict and original growth formulations should be distinguished rather than asserted equivalent. Exponentials with algebraic exponent satisfy the stronger conditions and a first-order differential equation.
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