×

zbMATH — the first resource for mathematics

On rational de Rham cohomology associated with the generalized Airy function. (English) Zbl 0887.33007
The classical Airy integral is known to be \(\int e^{tx-t^3/3} dt\). Its generalization to several variables was given by I. M. Gel’fand, V. S. Retakh and V. V. Serganova [Sov. Math., Dokl. 37, No. 1, 8-12 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 1, 17-21 (1988; Zbl 0699.33012)]. An important related problem is a cohomological interpretation of the integral, i.e., to clarify the structure of the cohomology group of a twisted complex associated to the integral. The author solves this problem in this paper.
Reviewer: T.Sasaki (Kobe)

MSC:
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] K. Aomoto , Les équation aux différences linéaires et les intégrales des functions multiformes , J. Fac. Sci. Univ. Tokyo , Sec. IA 22 ( 1975 ), 271 - 297 . Zbl 0339.35021 · Zbl 0339.35021
[2] V. Arnold - A. Varchenko - S. Goussein-Zadé , Singularité des applications différentiables , Vol. 1 - 2 , Mir , Moscow , 1986 .
[3] Godement , Topologie algébrique et théorie des faisceaux , Herman , Paris , 1958 . Zbl 0080.16201 · Zbl 0080.16201
[4] I.M. Gelfand - V.S. Retahk , V.V. Serganova , GeneralizedAiryfunctions, Schubert cells, and Jordan groups , Dokl. Akad. Nauk. SSSR 298 ( 1988 ), 17 - 21 ; English transl. in Soviet Math. Dokl . 37 ( 1988 ), 8 - 12 . Zbl 0699.33012 · Zbl 0699.33012
[5] Y. Haraoka - H. Kimura , Contiguity relations of the generalized confluent hypergeometric functions , Proc. Japan Acad . 69 ( 1993 ), 105 - 110 . Article | Zbl 0812.33007 · Zbl 0812.33007 · doi:10.3792/pjaa.69.105 · minidml.mathdoc.fr
[6] K. Iwasaki - H. Kimura - S. Shimomura - M. Yoshida , From Gauss to Painlevé , Vieweg , Wiesbaden , 1991 . Zbl 0743.34014 · Zbl 0743.34014
[7] H. Kimura , The degeneration of the two dimensional Garnier system and the polynomial Hamiltonian structure , Ann. Mat. Pura App . 155 ( 1989 ), 25 - 74 . Zbl 0693.34043 · Zbl 0693.34043 · doi:10.1007/BF01765933
[8] H. Kimura , On rational de Rham cohomology associated with the generalized confluent hypergeometric functions I, P1 case , to appear in Royal Society of Edimburgh . Zbl 0883.33009 · Zbl 0883.33009 · doi:10.1017/S0308210500023544
[9] H. Kimura , On Wronskian determinants of the generalized confluent hypergeometric functions , ( 1994 ). preprint.
[10] H. Kimura - Y. Haraoka - K. Takano , The generalized confluent hypergeometric functions , Proc. Japan Acad . 68 ( 1992 ), 290 - 295 . Article | Zbl 0773.33004 · Zbl 0773.33004 · doi:10.3792/pjaa.68.290 · minidml.mathdoc.fr
[11] H. Kimura - Y. Haraoka - K. Takano , On confluences of the general hypergeometric systems , Proc. Japan Acad . 69 ( 1993 ), 99 - 104 . Article | Zbl 0822.33007 · Zbl 0822.33007 · doi:10.3792/pjaa.69.99 · minidml.mathdoc.fr
[12] H. Kimura - Y. Haraoka - K. Takano , On contiguity relations of the confluent hypergeometric systems , Proc. Japan Acad . 70 ( 1994 ), 47 - 49 . Article | Zbl 0798.33009 · Zbl 0798.33009 · doi:10.3792/pjaa.70.47 · minidml.mathdoc.fr
[13] H. Kimura - T. Koitabashi , Normalizer of maximal abelian subgroups of GL(n) confluent hypergeometric functions , Kumamoto J. Math. 9 ( 1996 ), 13 - 43 . Zbl 0848.33009 · Zbl 0848.33009
[14] I.G. Macdonald , Symmetric functions and Hall polynomials , Oxford University Press , Oxford , 1979 . Zbl 0487.20007 · Zbl 0487.20007
[15] H. Matsumura , Commutative algebra, second edition , Benjamin , 1980 . Zbl 0441.13001 · Zbl 0441.13001
[16] M. Noumi , Expansion of the solutions of a Gauss-Manin system at a point of infinity , Tokyo J. Math. 7 ( 1984 ), 1 - 60 . Zbl 0554.32009 · Zbl 0554.32009 · doi:10.3836/tjm/1270152991
[17] K. Okamoto , Isomonodromic deformation and Painlevé equations and the Garnier system , J. Fac. Sci. Univ. Tokyo Sec. IA, Math. 33 ( 1986 ), 576 - 618 . Zbl 0631.34011 · Zbl 0631.34011
[18] K. Okamoto - H. Kimura , On particular solutions of Garnier systems and the hypergeometric functions of several variables , Quarterly J. Math. 37 ( 1986 ), 61 - 80 . Zbl 0597.35114 · Zbl 0597.35114 · doi:10.1093/qmath/37.1.61
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.