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Uspekhi Mat. Nauk, 2001, Volume 56, Issue 4(340), Pages 3–34 (Mi umn414)  

This article is cited in 2 scientific papers (total in 3 papers)

$\mathscr N$-functions and their relationship with solutions of general hypergeometric systems and $GG$-systems

I. M. Gel'fand, M. I. Graev

Scientific Research Institute for System Studies of RAS

Abstract: A function $\mathscr N(z,x,\omega)$ on $\mathbb C^n\times\mathbb C^N$ is assigned to any non-singular $n\times N$ complex matrix $\omega$, where $n$ and $N\geqslant n$ are arbitrary positive integers. A relationship is established between these functions and the solutions of general hypergeometric systems of differential equations and their generalizations, the so-called $GG$-systems. It is natural to treat the functions $\mathscr N(z,x,\omega)$ as regularizations of solutions of these systems. Conversely, from any function $\mathscr N(z,x,\omega)$ one can recover the set of solutions of the corresponding $GG$-system. Also considered are analogues of $GG$-systems and related functions $\mathscr N(z,x,\omega)$ obtained by replacing the differentiation operators $\partial/\partial x_j$ by operators of more general form, in particular, by $q$-differentiation operators.

DOI: https://doi.org/10.4213/rm414

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English version:
Russian Mathematical Surveys, 2001, 56:4, 615–647

Bibliographic databases:

UDC: 517.58
MSC: Primary 33C70; Secondary 46F12, 34B30
Received: 04.07.2001

Citation: I. M. Gel'fand, M. I. Graev, “$\mathscr N$-functions and their relationship with solutions of general hypergeometric systems and $GG$-systems”, Uspekhi Mat. Nauk, 56:4(340) (2001), 3–34; Russian Math. Surveys, 56:4 (2001), 615–647

Citation in format AMSBIB
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\jour Uspekhi Mat. Nauk
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\jour Russian Math. Surveys
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\pages 615--647
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    This publication is cited in the following articles:
    1. I. M. Gel'fand, M. I. Graev, V. S. Retakh, “Hypergeometric functions over an arbitrary field”, Russian Math. Surveys, 59:5 (2004), 831–905  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. A. M. Vershik, I. M. Gel'fand, S. G. Gindikin, A. A. Kirillov, G. L. Litvinov, V. F. Molchanov, Yu. A. Neretin, V. S. Retakh, “Mark Iosifovich Graev (to his 85th brithday)”, Russian Math. Surveys, 63:1 (2008), 173–188  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Ochiai H., Zunderiya U., “A Generalized Hypergeometric System”, J. Math. Sci.-Univ. Tokyo, 20:2 (2013), 285–315  mathscinet  zmath  isi
  • Успехи математических наук Russian Mathematical Surveys
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