General hypergeometric systems of equations and series of hypergeometric type.

*(English. Russian original)*Zbl 0798.33010
Russ. Math. Surv. 47, No. 4, 1-88 (1992); translation from Usp. Mat. Nauk 47, No. 4(286), 3-82 (1992).

This is a survey of a series of works of Gel’fand and his collaborators on hypergeometric functions (HGF). The paper consists of two parts. In the first part the authors introduce hypergeometric series called \(\Gamma\)-series and the holonomic system satisfied by them. Let \(B\subset \mathbb{Z}^ N\) be a lattice, \(\gamma\in \mathbb{C}^ N\). The \(\Gamma\)-series is \(F_ B(\gamma,x)= \sum_{b\in B} (\prod_{1\leq i\leq N} {{x_ i^{\gamma_ i+ b_ i}} \over {\Gamma(\gamma_ i+ b_ i+1)}} )\). Under certain condition on \(B\) and a parameter \(\gamma\) they showed the convergence of the series. They also discussed the relation with the Mellin-Sato hypergeometric series. Let \(L\subset \mathbb{C}^ N\) be the subspace spanned by \(B\). Then the holonomic system is
\[
\begin{aligned} \sum a_ i x_ i {{\partial\phi} \over {\partial x_ i}} &= \langle a,\alpha\rangle\phi \qquad \text{for any } a\in \text{ann }L,\\ \prod_{i:b_ i>0} \Bigl( {\partial \over {\partial x_ i}} \Bigr)^{b_ i} \phi &= \prod_{i: b_ i<0} \Bigl( {\partial \over {\partial x_ i}} \Bigr)^{-b_ i} \phi, \qquad b\in B,\end{aligned}
\]
where \(\alpha\) is the image of \(\gamma\) by the projection \(\mathbb{C}^ N\to \mathbb{C}^ N/L\). Along the same idea, the authors introduce difference analogue and \(q\)-analogue of the above series and a system.

In the second part, a general hypergeometric function (HGF) connected with the Grassmannian \(G_{k,n}\) is treated. It is regarded as a special case of that introduced in the first part, but this was the starting point of a series of work. HGF is defined as a Radon transform of a character of Cartan subgroup of \(GL(n)\). This formulation generalizes the Euler integral representation for the Gauss hypergeometric function. For HGF, the authors study the holonomic system, series solution, the restriction formula of HGF to the singular locus of the system (called stratum). A generalization of Airy function is also treated briefly.

In the second part, a general hypergeometric function (HGF) connected with the Grassmannian \(G_{k,n}\) is treated. It is regarded as a special case of that introduced in the first part, but this was the starting point of a series of work. HGF is defined as a Radon transform of a character of Cartan subgroup of \(GL(n)\). This formulation generalizes the Euler integral representation for the Gauss hypergeometric function. For HGF, the authors study the holonomic system, series solution, the restriction formula of HGF to the singular locus of the system (called stratum). A generalization of Airy function is also treated briefly.

Reviewer: H.Kimura

##### MSC:

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |