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Generalized Airy functions, Schubert cells, and Jordan groups. (English. Russian original) Zbl 0699.33012
Sov. Math., Dokl. 37, No. 1, 8-12 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 1, 17-21 (1988).
The authors develop an approach to the theory of special functions of several variables founded by I. M. Gel’fand in connection with generalized hypergeometric functions [Dokl. Akad. Nauk SSSR 288, 14-18 (1986; Zbl 0645.33010)]. This paper deals with the generalization of the theory of Airy functions to the case of several variables. For this functions the system of differential equations is defined on a manifold \(Z^ n_ k\) of rectangular matrices \((z_{ij})\), \(i=1,...,k\), \(j=1,...,n\). It is shown that this system is holonomic, and hence has a finite number of linearly independent solutions in a neighborhood of each point of \(Z^ n_ k\). The formula for the dimension of the space of solutions on a certain open everywhere dense set \(W\subset Z^ n_ k\) (the so-called Schubert cell) is given. For the system rewritten in coordinates transversal to the trajectories of the group GL(k) a set of linearly independent solutions is constructed.
Reviewer: A.V.Rozenblyum

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