Value distribution of a pair of meromorphic functions

The well-known Picard theorem shows only what happens on the Picard exceptional value of a meromorphic function f. In this paper, we mainly consider what happens on the common or different Picard exceptional values of two or three meromorphic functions with certain types. For instance, we will provide some new results and discussions for the generalized Picard exceptional values or small functions of a pair of meromorphic functions $f^n{g}^{\left(k\right)}$ and $g^m{f}^{\left(k\right)}$, these two functions are the crossed variants of the complex differential polynomials in Hayman's conjecture, where n, m are positive integers and $k\ge 0$. We give more details on the case that $f{g}^{\prime }$ and $g{f}^{\prime }$ when f and g are exponential polynomials in particular.