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

TL;DR

This paper extends classical results on the number of linearly independent solutions of linear differential and difference equations, providing refined growth estimates and conditions for solutions of infinite order, including in the unit disc and for q-difference equations.

Contribution

It generalizes Frei's theorem by allowing transcendental coefficients and establishing new growth bounds, increasing the count of solutions of infinite order.

Findings

At least n-p solutions of infinite order exist under generalized conditions.

Zero is the only finite deficient value for these solutions.

Results apply to differential, difference, and q-difference equations.

Abstract

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.