E-function ($f(z)={\sum }_{n\ge 0}{a}_n{z}^n/n!$)

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 $d_n\le C^{n+1}$ 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.

Work in the compositum of the two coefficient fields. Conjugate coefficient bounds add, while multiplying the common denominators clears both coefficient lists. For Siegel's original bounds $O_{ϵ}(n^{ϵn})$, use $ϵ/2$ in each denominator estimate. A nonzero linear differential equation for the sum follows because its derivatives lie in the sum of the two finite-dimensional derivative spaces over the rational function field. The stricter exponential arithmetic bounds are preserved as well.

Let a vector of E-functions satisfy $f^{\prime }=A(z)f$ with $A\in {\mathrm{Mat}}_s(\overline{\mathbb{Q}}(z))$. At a nonzero algebraic point $\xi$ which is not a pole of $A$, the transcendence degree of the values over $\overline{\mathbb{Q}}$ equals the transcendence degree of the functions over $\overline{\mathbb{Q}}(z)$. In particular algebraically independent functions give algebraically independent values. The exponential system $f_j=e^{{\gamma }_{j}z}$, for rationally independent algebraic ${\gamma }_j$, recovers the Lindemann–Weierstrass theorem by evaluating at one.