A classical theorem of Frei states that if $A_p$ is the last transcendental function in the sequence $A_0,\ldots ,A_{n-1}$ of entire functions, then 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.

Frei 的一个经典定理指出，如果 $A_p$ 是整个函数序列 $A_0,\ldots ,A_{n-1}$ 中的最后一个超越 函数，那么微分方程 $f^{(n )}+A_{n-1}f^{(n-1)}+\cdots +A_{1}f'+A_{0}f=0$ 包含至少 $np$ 整个无限阶函数。这里，超越系数 $A_p$ 支配多项式系数 $A_{p+1},\ldots ,A_{n-1}$ 的增长 。通过以不同方式表达 $A_p$ 的支配地位 并允许系数 $A_{p+1},\ldots ,A_{n-1}$ 为了超越，我们证明 Frei 定理的结论与对解增长的渐近下界的额外估计仍然成立。有时，这些新的精炼结果给出了比 Frei 的原始定理更多的无限阶线性无关解。对于此类解决方案，我们表明 $0$ 是唯一可能的有限缺陷值。以前，已知此属性适用于所谓的可容许解，并且通常被引用为 Wittich 定理。讨论了单位圆盘中线性微分方程以及复差分和复q差分方程的类似结果。