A General Characterization on the Uniqueness Problem of L-Functions and General Meromorphic Functions

In the paper, concerning a question of Yi [23], we study general criterion for the uniqueness of an L-function and a general meromorphic function. Our results improve and extend all the existing results in this direction [23, 18, 17, 4] to the most general setting. Moreover, we have exhibited a handsome number of examples to justify our claims as well as to confirm the wide-ranging applications of our results.

💡 Research Summary

The paper addresses a fundamental question in value‑distribution theory: under what conditions does sharing a set of values force an L‑function and a general meromorphic function to be identical? Earlier works (Theorems A–F) dealt only with very specific polynomials—namely (P(z)=z^{n}+az^{m}+b) or (P(z)=z^{n}+az^{,n-m}+b)—and required the meromorphic function to have only finitely many poles. Moreover, those results imposed relatively strong degree constraints (e.g., (n\ge 2m+5) or (n\ge 2m+11) for IM sharing) and often assumed simple zeros of the polynomial. The present work removes all these restrictions and provides a unified, far‑reaching framework.

Main contributions

1. General polynomial framework.
   The authors introduce an arbitrary polynomial of degree (n), \