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

• Published in: Canadian journal of mathematics Vol. 73; no. 6; pp. 1556 - 1591
• Main Authors: Heittokangas, Janne, Yu, Hui, Zemirni, Mohamed Amine
• Format: Journal Article
• Language: English
• Published: Canada Canadian Mathematical Society 01.12.2021; Cambridge University Press
• Subjects: Coefficients; Difference equations; Differential equations; Entire functions; Lower bounds; Mathematical analysis; Polynomials; Theorems; Transcendental functions; Deficient values; Frei’s theorem; Wittich’s theorem; entire functions; linear difference equations; 34M10; 30D35; linear differential equations
• ISSN: 0008-414X; 1496-4279
• DOI: 10.4153/S0008414X20000607

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 in...