Unicity Results Concerning of difference monomials of L-function and a meromorphic function

Abstract

In this paper, we study the value distribution of $\mathcal{L}$-function in the extend Selberg class and a non-constant transcendental meromorphic $\mathsf{f}$ function with finitely many zeros of finite order, sharing a polynomial with its difference monomial. Meromorphic functions and $L$-functions are central to Nevanlinna theory, a branch of complex analysis focusing on the distribution of zeros and poles of analytic functions. The  Meromorphic functions are unique due to their combination of meromorphicity and analyticity. They are defined on the entire complex plane except for isolated poles in the complex analysis. On the other hand, $L$-functions, arising notably in number theory and automorphic forms, exhibit unique properties related to their zeros and poles. One classic example of a prime number distribution is the Riemann $\zeta$-function. In this paper, we analyse the uniqueness results between a non-constant meromorphic function $\mathsf{f}$ having finitely many zeros and $L$-function, when their difference monomial $\mathsf{f}^{n}\prod\limits_{j=1}^{s}\mathsf{f}(z+c_{j})^{\mu_{j}}$ and $\mathcal{L}^{n}\prod\limits_{j=1}^{s}\mathcal{L}(z+c_{j})^{\mu_{j}}$ share a non-zero polynomial $p$. Our results extends and improves the earlier results of Harina P. Waghamore and Manjunatha B E\cite{bib6}.