Abstract
Recently, C.-.C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form f n + L(z, f) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 +q(z)f(z +1) = p(z), where p(z),q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ ℂ, equations of the form f(z)n +q(z)eQ(z) f(z +c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.
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The first author is supported by the China Scholarship Council (CSC); the third author is supported in part by the Academy of Finland #121281
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Wen, Z.T., Heittokangas, J. & Lain, I. Exponential polynomials as solutions of certain nonlinear difference equations. Acta. Math. Sin.-English Ser. 28, 1295–1306 (2012). https://doi.org/10.1007/s10114-012-1484-2
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DOI: https://doi.org/10.1007/s10114-012-1484-2