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The generalized confluent hypergeometric functions. (English) Zbl 0773.33004
This article extends the theory of generalized hypergeometric functions of Gelfand and others to those including the case corresponding the classical confluent hypergeometric functions. Let $$Z_{r,n}$$ be the set of $$r\times n$$ complex matrices of maximal rank. Let $$\lambda=(\lambda_ 1,\dots,\lambda_ \ell)$$, $$\lambda_ 1+\dots+\lambda_ \ell=n$$ be a partition. Let $$H_ \lambda:=J(\lambda_ 1)\times \dots\times J(\lambda_ \ell)$$ denote the maximal commutative subgroup of $$GL(n,\mathbb{C})$$ associated with the Young diagram defined by $$\lambda$$. Let $$\chi_ \alpha$$ be a character of the universal covering group of $$H_ \lambda$$. Then a generalized confluent hypergeometric function $$F(z)$$ on $$Z_{r,n}$$ is defined as one satisfying \begin{alignedat}{2} F(zc) &= F(z)\chi_ \alpha(c) \quad &&\text{for } c\in H_ \lambda,\\ F(gz) &= (\text{det }g)^{-1} F(z) \quad &&\text{for } g\in GL(r,\mathbb{C}).\end{alignedat} For $$\lambda=(1,\dots,1)$$ this reduces to Gelfand’s generalized hypergeometric function, and for $$\lambda=(n)$$ to the generalized Airy function of Gelfand-Retahk-Serganova. The holonomicity, integral representation, symmetries, relations with the classical confluent hypergeometric functions are discussed.

##### MSC:
 33C15 Confluent hypergeometric functions, Whittaker functions, $${}_1F_1$$ 33C70 Other hypergeometric functions and integrals in several variables 35N99 Overdetermined problems for partial differential equations and systems of partial differential equations 32C38 Sheaves of differential operators and their modules, $$D$$-modules 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
##### Keywords:
Grassmannian manifolds
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##### References:
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