Forward Differences of Entire Solutions of Second-Order Complex Differential Equations

Abstract

In this paper the effect of the translation of the coefficient by a fixed non-zero amount on the forward differences of entire solutions of second order complex differential equations with entire coefficients is examined. The basic coefficient-difference identity implies that the absence of shift periodicity is the same forcing as provided by translation. If the coefficient induced by a nontrivial forward difference, the full local principal part at each zero is determined and an exact criterion is given for when the induced meromorphic coefficient is an entire function. Coefficient periodicity and preservation of the original homogeneous equation is then described by a shift-Wronskian evolution identity. A nonconstant polynomial exponent is proven to be linear, and for exponential coefficients, the shift preservation condition is simplified to a discrete resonance condition. For the periodic case, classical Floquet translation results in a second-order difference relation, a parabolic-trace consequence and a zero-transfer classification of solutions and higher forward differences. Examples of constant-coefficient, exponential-Bessel, Airy and quadratic-exponential are used to illustrate the theory. Numerical examples of zero-counting, monodromy computation, and power-series residuals are only provided for the purpose of performing reproductions and checks of the exact analytic results.