The space $\mathscr E_\rho$ of the entire functions of order $\rho$ ( $1<\rho<\infty$ ) with the usual topology and the operator $\mathscr L$ , induced by a differential operation $l[y]=y^n+p_{n-2}(z)y^{n-2}+\dots+p_0(z)y$ , $n>1$ , and “boundary” conditions $F_i[y]=0$ ( $i=1,\dots,n$ ), where the $F_i$ are linear functionals on $\mathscr E_\rho$ . Conditions are indicated under which the formal expansion $f\sim-\Sigma_\lambda\operatorname{Res}(\mathscr L-\lambda E)^{-1}f$ uniquely determines an element $f\in\mathscr E_\rho$ . As a corollary it is established that if $\Delta(\lambda)=\Sigma c_k\lambda^k\in\mathscr E_\mu$, $\mu>1$ , has an infinite number of zeros and $f(z)\in\mathscr E_\rho$ , $\rho<\mu(\mu-1)$ , then $f(z)\equiv0$ whenever $$ \sum^\infty_{k=1}\frac{c_k(\lambda^{k-1}f(0)+\dots+f^{(k-1)}(0))}{\Delta(\lambda)} $$is an entire function.Bibliography: 10 titles.