Geodesic
$\mathscr N$-functions and their relationship with solutions of general hypergeometric systems and $GG$-systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 56 (2001) no. 4, pp. 615-647

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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. 
@article{RM_2001_56_4_a0,
     author = {I. M. Gel'fand and M. I. Graev},
     title = {$\mathscr N$-functions and their relationship with solutions of general hypergeometric systems and $GG$-systems},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {615--647},
     publisher = {mathdoc},
     volume = {56},
     number = {4},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2001_56_4_a0/}
}
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I. M. Gel'fand; M. I. Graev. $\mathscr N$-functions and their relationship with solutions of general hypergeometric systems and $GG$-systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 56 (2001) no. 4, pp. 615-647. http://geodesic.mathdoc.fr/item/RM_2001_56_4_a0/