Uniqueness of Q-difference of meromorphic functions sharing a small function with finite weight

In this paper, we study the value distribution and uniqueness of linear -difference polynomial and its -difference operator , for a transcendental meromorphic function having zero order. Our results extends, improves and generalizes some of the earlier results due to Thin [Thin, N. V. Uniqueness of meromorphic functions and -difference polynomials sharing small functions.  Bull. Iranian Math. Soc. 43 (2017), no.~3, 629–647]; Dyavanal and Desai [Dyavanal, R. S.; Desai, R. V. Uniqueness of -shift difference polynomials of meromorphic functions sharing a small function.  Math. Bohem. 145 (2020), no.~3, 241–253].