Some algebraic differential equations with few transcendental solutions (Q2348492)

\textit{H. Wittich} [Math. Ann. 122, 221--234 (1950; Zbl 0038.23001)] showed that the differential equation \[ f^{(k)}= f^{(j_1)}\cdots f^{(j_d)}\text{ with }d>1\text{ and each }j_1< k\tag{\(*\)} \] has no entire transcendental solutions. The present, interesting paper studies meromorphic solutions in \(\mathbb{C}\) of autonomous algebraic differential equations of the form \((*)\). Using a theorem of \textit{W. K. Hayman} and \textit{J. Miles} [Complex Variables, Theory Appl. 12, No. 1--4, 245--260 (1989; Zbl 0643.30021)] a short proof is provided for the statement that there are no meromorphic solutions of \((*)\) with \(\infty\) as a Nevanlinna exceptional value (proved earlier by \textit{J.-J. Zhang} and \textit{L.-W. Liao} [J. Math. Anal. Appl. 397, No. 1, 225--232 (2013; Zbl 1270.34201)]). A second theorem proved is that each periodic meromorphic solution of \((*)\) with only finitely many poles in a period strip has the form \(f(z)= g(e^{cz})\) with \(g\) rational and \(c\) a constant. The latter theorem relies on Theorems 2.1 and 2.2 by \textit{W. Bergweiler} et al. [Proc. Lond. Math. Soc. (3) 97, No. 2, 368--400 (2008; Zbl 1174.37011)] in a manner shown by \textit{A. E. Eremenko} et al. [Math. Proc. Camb. Philos. Soc. 146, No. 1, 197--206 (2009; Zbl 1163.34058)], and the fact that the multiplicity \(m\) of the pole in any formal Laurent series solution of \((*)\) must satisfy \(k= m(d-1)+ h\), where \(h= J_1+\cdots+ j_d\). The remaining part of the paper employs the above to discover the special shape for the Laurent series solutions of \((*)\) when a meromorphic solution of \((*)\) has at least one pole of multiplicity \(m\).