Dual exponential polynomials and linear differential equations

Abstract

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

Quantitative Properties of Meromorphic Solutions to Some Differential-Difference Equations

“…On the other hand, if P 1 e P 0 ≡ 0 in (1.1), Voorhoeve et al [15] observed in 1975 that every exponential polynomial is a solution to this new equation. In addition, when the coefficients of this new equation are exponential polynomials and exactly one coefficient has order strictly larger than those of the others, Wen et al [16] recently proved that a transcendental exponential polynomial solution to such an equation has a specific dual relation to the maximum order coefficient.…”

“…Motivated by the consideration of transcendental exponential polynomials as in [15,16], we also discuss exponential polynomial solutions to the equation…”

Exponential Polynomials and Nonlinear Differential-Difference Equations

“…holds for an exponential polynomial f(z) in form (23) (also see [21], Section 3). Some auxiliary results are necessary.…”

Dual exponential polynomials and a problem of Ozawa

“…It was discovered in [18,Lemma 1] that in this representation one has C j = 0 for 1 ≤ j ≤ m. Substituting the subnormal solution f into (1.1), we get C j e jz = m 2 .…”