## $$GG$$-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)

### MSC:

 33C70 Other hypergeometric functions and integrals in several variables 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33C05 Classical hypergeometric functions, $${}_2F_1$$
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