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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.

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
Full Text: DOI
[1] I. M. Gelfand : General theory of hypergeometric functions. Dokl. Akad. Nauk. SSSR., 288, 14-48 (1986); English translation in Soviet Math. Dokl, 33, 9-13 (1986). · Zbl 0645.33010
[2] I. M. Gelfand and S. I. Gelfand : Generalized hypergeometric equations, ibid., 288, 289-283 (1986); 33, 643-646 (1986).
[3] I. M. Gelfand, V. S. Retakh and V. V. Serganove: Generalized Airy functions, Schublrt cells, and Jordan groups, ibid., 298, 17-21 (1988); ibid., 37, 8-12 (1988). · Zbl 0699.33012
[4] K. Iwasaki et at.: From Gauss to Painleve. Virweg, Wiesbaden (1991).
[5] H. Kimura: The degeneration of the two dimensional Garnier system and polynomial Hamiltonian structure. Ann. Mat. Pura Appl., 155, 25-74 (1989). · Zbl 0693.34043 · doi:10.1007/BF01765933
[6] K. Okamoto: Isomonodromic defomation and Painleve equations and the Garnier system. J. Fac. Sci. Univ. Tokyo, Sec. IA, 33, 576-618 (1986). · Zbl 0631.34011
[7] K. Okamoto and H. Kimura : On particular solutions of Garnier systems and the hypergeometric functions of several variables. Quarterly J. Math., 37, 61-80 (1986). · Zbl 0597.35114 · doi:10.1093/qmath/37.1.61
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