Complex analogs of the Gelfand–Tsetlin patterns are introduced. Infinite-dimensional representations of \(\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)\) in the vector spaces spanned on these patterns are constructed. Exponentials of these representations are described. These exponentials are operators T(x), x∈GL(n,C), defined only in neighborhoods of the identity element of GL(n,C). A system of differential-difference equations for matrix elements of operators T(x) is constructed. Explicit formulas for matrix elements are obtained for the case x∈Z±, where Z+ and Z− are the triangular unipotent subgroups. Representations of \(\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)\) are also constructed; bases of these representations consist of Gelfand–Tsetlin patterns having infinitely many rows.