$\mathscr N$-functions and their relationship with solutions of general hypergeometric systems and $GG$-systems

A function $\mathscr N(z,x,\omega)$ on $\mathbb C^n\times\mathbb C^N$ is assigned to any non-singular $n\times N$ complex matrix $\omega$, where $n$ and $N\geqslant n$ are arbitrary positive integers. A relationship is established between these functions and the solutions of general hypergeometric systems of differential equations and their generalizations, the so-called $GG$-systems. It is natural to treat the functions $\mathscr N(z,x,\omega)$ as regularizations of solutions of these systems. Conversely, from any function $\mathscr N(z,x,\omega)$ one can recover the set of solutions of the corresponding $GG$-system. Also considered are analogues of $GG$-systems and related functions $\mathscr N(z,x,\omega)$ obtained by replacing the differentiation operators $\partial/\partial x_j$ by operators of more general form, in particular, by $q$-differentiation operators.