On the hypertranscendence of meromorphic solutions of certain functional equations

1979 ◽  
Vol 83 (1-2) ◽  
pp. 45-54 ◽  
Author(s):  
Steven B. Bank
Keyword(s):  
Differential Equations ◽  
Functional Equations ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Algebraic Differential Equations

SynopsisThis paper deals mainly with showing that meromorphic solutions of Poincaré functional equations cannot satisfy algebraic differential equations. Results of this type for entire solutions have previously been obtained by H. Wittich.

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

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