E-function

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 $d_n\le C^{n+1}$ making $d_n{a}_0,\dots ,d_n{a}_n$ algebraic integers. Siegel's original arithmetic growth condition permits, for each $ϵ>0$, bounds of size $O_{ϵ}(n^{ϵn})$ 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.