On the representation of arbitrary functions of two complex variables by double functional series of Dirichlet type

We investigate the question of representing a function $F(z,s)$ by a functional series of the form
\begin{equation} \sum^\infty_{n,k=1}a_{nk}A(z,s,\lambda_n,\mu_k), \tag{1} \end{equation}
where $A(z,s,\lambda,\mu)$ is a function of sufficiently general character. We establish a rule by which an arbitrary function $F(z,s)$ can be put into correspondence with a series of the form (1), and also establish a formula for the difference between $F(z,s)$ and a partial sum of the series (1).