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**\(GG\)-functions and their relation to general hypergeometric functions.**
*(English.
Russian original)*
Zbl 0922.33010

Russ. Math. Surv. 52, No. 4, 639-684 (1997); translation from Usp. Mat. Nauk 52, No. 4, 3-48 (1997).

The aim of this paper is to give a new approach to hypergeometric functions. According to this approach, hypergeometric functions are defined as solutions of GG-systems which are collections of linear relations for a function \(f(a,\alpha)\) of \(a,\alpha \in {\mathbb C}^N\), its first partial derivative with respect to \(a_i\) and shifts with respect to \(a_i\). A GG-system is determined by an arbitrary linear subspace \(L\subset {\mathbb C}^N\) and consists of equations \((\partial _i f)(\alpha ,a)=f(\alpha -e_i ,a)\), \(i=1,\cdots ,N\), \(f(\alpha +l,a)=f(\alpha ,a)\), \(l\in L\), \(\sum _{i=1}^N \omega _i (\theta _i f)(\alpha ,a)=\langle \omega ,\alpha\rangle f(\alpha ,a)\), \(\omega =(\omega _1,\cdots ,\omega _N)\in ({\mathbb C}^N)'\), \(\omega \bot L\). Here \(\partial _i =\partial /\partial a_i\), \(\theta _i =a_i \partial /\partial a_i\), and \(\{ e_i\}\) is the standard basis in \({\mathbb C}^N\). Solutions of this system, holomorphic in \(a\), are called GG-functions associated with the subspace \(L\). If \(L\) is linearly generated by a sublattice \(\Lambda \subset {\mathbb Z}^N\), then any GG-function associated with \(L\) is a general hypergeometric function associated with \(\Lambda\). If the subspace \(L\) is not generated by a sublattice \(\Lambda \subset {\mathbb Z}^N\), then the GG-functions associated with \(L\) do not reduce to the solutions of differential equations. Thus, the class of GG-functions is larger than the class of general hypergeometric functions. The authors give a description of the solutions of the GG-system associated with an arbitrary subspace \(L\). The description is presented both in the form of series and in the integral form, and it does not depend on the structure of \(L\). If \(L\) is generated by a sublattice \(\Lambda \subset {\mathbb Z}^N\), then new bases are constructed in the space of solutions of the corresponding hypergeometric system of equations. The paper consists of two chapters. The first chapter is devoted to GG-functions associated with one-dimensional subspaces \(L\). In this case GG-functions reduce to functions of one variable when regarded as functions of \(a_i\). Their relations with the known hypergeometric functions of one variables (the Pochhammer functions \({}_p F_q\), the Meijer \(G\)-functions \(G^{mn}_{pq}\), and the Fox \(H\)-functions \(H^{mn}_{pq}\)) are established. The second chapter is devoted to studying the “resonance” case. It is proved that for certain GG-systems the restrictions of the GG-functions to suitable planes in the parameter space satisfy additional differential equations. In particular, it is shown that the hypergeometric system of differential equations associated with the Grassmannian \(G_{p,n}\) of \(p\)-dimensional subspaces of \(n\)-dimensional complex space arises as the resonance case of the GG-system associated with the space \({\mathbb C}^p\otimes {\mathbb C}^n\).

Reviewer: A.Klimyk (Kyïv)