In this research article, we investigates the value distribution of linear q-dierence
operators Lk(f;
q;c) and Lk(g;
q;c), for a transcendental entire functions of zero order. At
the same time we also investigate the uniqueness problems when two linear q- dierence
operators of entire functions share one value with nite weight. Our results extends the
previous theorems of existing studies [11],[5].
In this paper, we assume that the reader is familiar with the fundamental results [7],[14],[15].
We adopt the standard notations of the Nevanlinna theory of meromorphic function m(r; f),
N(r; f), N(r; 0; f) and T(r; f) denote the proximity function, the counting function, the
reduced counting function and the characteristic function of f(z), respectively.
Let f and g be two non-constant meromorphic functions dened in the complex plane and
S(r; f) denote any quantity satisfying S(r; f) = o(T(r; f)) as r ! 1 possibly exceptional
set of nite linear measure. A meromorphic function (6 0;1) is called a small function
with respect to f, if T(r; ) = S(r; f). If for some a 2 C [1, the zeros of f 􀀀 a and g 􀀀 a
coincide in locations and multiplicity, we say that f and g share the value a CM(Counting
Multiplicities). On the other hand, if the zeros of f 􀀀 a and g 􀀀 a coincide only in their
locations, then we say that f and g share the value a IM(Ignoring Multiplicities).