On meromorphic solutions of certain Fermat-type difference and analogues equations concerning open problems
Rajib Mandal, Raju Biswas, Sudip Kumar Guin
TL;DR
This paper studies non-constant meromorphic solutions of Fermat-type difference equations and shows such solutions are generated by Riccati-type meromorphic functions: every solution of $f^2(z)+f^2(z+c)=1$, $f^2(z)+f^2(qz)=1$, and $f^2(z)+f^2(qz+c)=1$ can be written as $f(z)=\frac{2\omega(z)}{1+\omega(z)^2}$ with $\omega$ satisfying a Riccati-type relation. The authors provide explicit examples, prove the main theorem by transforming to a Riccati form and applying Hadamard factorization and Nevanlinna theory, and address Liu–Yang open problems on derivative-type and shift equations by giving a complete description of transcendental entire and meromorphic solutions, including infinite zeros. Overall, the work offers a unified Riccati-based construction for Fermat-type equations in the difference setting and expands the catalog of known meromorphic solutions.
Abstract
In this paper, we have found that some certain Fermat-type shift and difference equations have the meromorphic solutions generated by Riccati type functions. Also we have solved the open problems posed by Liu and Yang (A note on meromorphic solutions of Fermat types equations, An. Stiint. Univ. Al. I. Cuza Lasi Mat. (N. S.), 62(2)(1), 317-325 (2016)). We have fortified the claims by some examples.