  Assuming that $A_0,\ldots,A_{n-1}$ are entire functions and that $p\in
\{0,\ldots,n-1\}$ is the smallest index such that $A_p$ is transcendental,
then, by a classical theorem of Frei, each solution base of the differential
equation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ contains at least
$n-p$ entire functions of infinite order. Here, the transcendental coefficient
$A_p$ dominates the growth of the polynomial coefficients
$A_{p+1},\ldots,A_{n-1}$. By expressing the dominance of $A_p$ in different
ways, and allowing the coefficients $A_{p+1},\ldots,A_{n-1}$ to be
transcendental, we show that the conclusion of Frei's theorem still holds along
with an additional estimation on the asymptotic lower bound for the growth of
solutions. At times these new refined results give a larger number of linearly
independent solutions of infinite order than the original theorem of Frei. For
such solutions, we show that $0$ is the only possible finite deficient value.
Previously this property has been known to hold for so-called admissible
solutions and is commonly cited as Wittich's theorem. Analogous results are
discussed for linear differential equations in the unit disc, as well as for
complex difference and complex $q$-difference equations.
