On the Study of Meromorphic Functions That Shares Small Functions Partially With the Second Order Difference Operator

Abstract

In this paper, we looked at some problems with the uniqueness of meromorphic functions with a second order difference operator. We looked at them from the point of view of partial sharing. We have obtained two uniqueness results. In the first theorem ${\mathrm{\Delta }}^2\mathfrak{g}\mathfrak{(}\mathfrak{z}\mathfrak{)}$ and $\mathfrak{g}\mathfrak{(}\mathfrak{z}\mathfrak{)}$ shares ${\mathfrak{a}}_1\mathfrak{(}\mathfrak{z}\mathfrak{)}$, ${\mathfrak{a}}_2\mathfrak{(}\mathfrak{z}\mathfrak{)}$, $\mathrm{\infty }$ CM, whereas in the second theorem $\mathfrak{g}\mathfrak{(}\mathfrak{z}\mathfrak{)}$ and ${\mathrm{\Delta }}^2\mathfrak{g}\mathfrak{(}\mathfrak{z}\mathfrak{)}$ partially share ${\mathfrak{a}}_1\mathfrak{(}\mathfrak{z}\mathfrak{)}$, ${\mathfrak{a}}_2\mathfrak{(}\mathfrak{z}\mathfrak{)}$ CM that generalizes the results due to Banerjee and Maity (Meromorphic function partially shares small functions or values with its linear c-shift operator, Bulletin of the Korean Mathematical Society 58(5) (2021), 1175 -- 1192), and Heittokangas et al., Uniqueness of meromorphic functions sharing values with their shifts, Complex Variables and Elliptic Equations 56(1-4) (2011), 81 -- 92.