Toward a unified theory special functions of several variables

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The objective of this project is to study the general hypergeometric functions (GHF) which were introduced by us to treat the classical special functions such as Gauss hypergeometric, Kummer's confluent hypergeometric, Bessel, Hermite and Airy function.1 : GHFs are defined as solutions of certain holonomic systems on the Grassmannian Gr_<r, n> and they have the integral representations in a formal sense. We try to understand these integrals in the framework of the de Rham theory, namely, as the dual pairing of cocycles and cycles of certain cohomology and homology groups. Here we treat this problem in the particular cases of GHF,the case of generalized Airy functions and the case of GHFs given by the one dimensional integrals.(1)In relation to the problem of expressing the holonomic system for the generalized Airy functions as the integrable holonomic connection outside of the singlar locus, we computed in [3] the cohomology group of the rational twisted de Rham complex associated with … More the representation. We showed that the cohomology groups vanish except for the r-th one and that dim H^r=_<n-2>C_<r-1>. Moreover we presented the conjecture that a basis of H^r is given in thems of the Schur functions.(2)We understand domains of integrations for the generalized Airy integrals as cycles of a homology group on P^r with the family of supports defined by the integrand. By using the r-dimensional saddle point method, we showed in [4] that the homology groups are trivial except for the r-th one and that r-th homology group forms a local system of Z-modules on the space of independent variables of the functions rank _<n-2>C_<r-1>.(3)In the case where the GHFs are given by the one dimensional integrals (in other terms, the confluent case of Lauricella's F_D), we showed that the rational de Rham cohomology groups are trival except for H^1, and gave a basis of H^1 explicitly.2 : It is known that the other special functions of confluent type are derived from the Gauss hypergeometric function by the limit processes called confluences. In [5] we showed that this phenomenon can be explained by the adjacency relations among the strata of the stratification naturally introduced in the set of regular elements in the Lie algebra gl_n. Furthermore we generalized the above limit process to GHF in general. Less