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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 differentialequation $f^{(n)}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ 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 differentways, and allowing the coefficients $A_{p+1},\ldots,A_{n-1}$ to betranscendental, we show that the conclusion of Frei's theorem still holds alongwith an additional estimation on the asymptotic lower bound for the growth ofsolutions. At times these new refined results give a larger number of linearlyindependent solutions of infinite order than the original theorem of Frei. Forsuch solutions, we show that $0$ is the only possible finite deficient value.Previously this property has been known to hold for so-called admissiblesolutions and is commonly cited as Wittich's theorem. Analogous results arediscussed for linear differential equations in the unit disc, as well as forcomplex difference and complex $q$-difference equations.

On the number of linearly independent rapid solutions to linear differential and linear difference equations