Unicity of transcendental meromorphic functions concerning differential-difference polynomials

Let $f$ and $g$ be two transcendental meromorphic functions of finite order with a Borel exceptional value $\mathrm{\infty }$, let $\alpha$ $(\not\equiv 0)$ be a small function of both $f$ and $g$, let $d,k,n,m$ and $v_j(j=1,2,\cdots ,d)$ be positive integers, and let $c_j(j=1,2,\cdots ,d)$ be distinct nonzero finite values. If $n\ge max\{2k+m+\sigma +5,\sigma +2d+3\}$, where $\sigma =v_1+v_2+\cdots +v_d$, and $(f^n(z)(f^m(z)-1)\prod _{j=1}^d{f}^{v_j}(z+c_j){)}^{(k)}$ and $(g^n(z)(g^m(z)-1)\prod _{j=1}^d{g}^{v_j}(z+c_j){)}^{(k)}$ share $\alpha$ CM then $f\equiv tg$, where $t^m=t^{n+\sigma }=1.$ This result extends and improves some restlts due to ^{[1,10,14,15,19]}.