Uniqueness of meromorphic functions concerning fixed points

In this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $f$ and $g$ be two nonconstant meromorphic functions, let $n,k$ be two positive integers with $n>3k+10.5-{\mathrm{\Theta }}_{min}(k+6.5)$, if ${\mathrm{\Theta }}_{min}\ge \frac{2.5}{k+6.5}$, otherwise $n>3k+8$, and let $(f^n{)}^{(k)}$ and $(g^n{)}^{(k)}$ share $z$ CM, $f$ and $g$ share $\mathrm{\infty }$ IM, then one of the following two cases holds: If $k=1$, then either $f(z)=c_1{e}^{c{z}^2}$, $g(z)=c_2{e}^{-c{z}^2}$, where $c_1,c_2$ and $c$ are three constants satisfying $4n^2(c_1{c}_2{)}^n{c}^2=-1$, or $f=tg$ for a constant $t$ such that $t^n=1$; if $k\ge 2$, then $f=tg$ for a constant $t$ such that $t^n=1$. Our results extend and improve some results due to ^[8,9,19,24].