Abstract
In this note we shall give a new formulation of Jackson integrals involved in basic hypergeometric functions through the classical Barnes representations. We define a q-analogue of de Rham cohomology which can be described by means of q-version of Sato’s b-functions and derive an associated holonomic q-difference system. The evaluation of its multiplicity will be given as the number of different asymptotic behaviours of Jackson integrals.
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References
Andrews, G., Problems and prospects for basic hypergeometric functions, The theory and applications of special functions, (R. Askey,ed.), Academic Press, New York, 1975, 191–224.
Aomoto, K., Les équations aux différences lineares et les intégrales des fonctions multiformes, J. of Fac. Sci. liniv. of Tokyo, 22(1975), 271–297.
Aomoto, K., A note on holonomic q-difference systems, Algebraic analysis I, (M. Kashiwara and T. Kawai, ed.), 1988, 25–28.
Aomoto, K., Finiteness of a cohomology associated with certain Jackson integrals, to appear in Tohoku J. of Math.
Aomoto, K., q-analogue of de Rham cohomology associated with Jackson integrals, I,II, to appear in Proc. of Japan Academy.
Askey, R., Ramanujan’s extensions of the gamma and beta functions, Amer. Math. Monthly, 87(1980), 346–359.
Bellardinelli, G., Fonctions hypergeometriques de plusieurs variables et resolutions analytiques des équations algébriques générales, Gauthiers-Vi11ars, 1960.
Bernshtein, D.N., The number of roots of a system of equations, Funct. Anal, and Its Appli., 9(1975), 185–185.
Danilov, V.I., The geometry of toric vaieties, Russ. Math. Surveys, 33(1978), 97–154.
Dantzig, G.B., Linear programming and extensions, Princeton Univ. Press, 1963.
Deligne, P., Equations différentielles, à points réguliers singuliers, Lec. Notes in Math., 163, Springer, 1970.
Gasper, G. and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, 1990.
Gustafson, R.A., A generalization of Selberg’s beta integral, Bull. A.M.S., 22(1990), 97–105.
Katz, N. and T. Oda, On the differentiation of the de Rham cohomology with respect to parameters, J. Math. Kyoto Univ., 8(1968), 199–213.
Kempf, G. et al, Toroidal embeddings I, Lee. Note in Math., 339, Springer, 1973.
Khovanskii, A.G., Newton polyhedra and toroidal varieties, Funct. Anal, and Its Appli., 11(1977), 56–57.
Kita, M. and M. Noumi, On the structure of cohomolgy groups attached to the integral of certain many-valued analytic functions, Japanese J. of Math., 9(1983), 113–157.
Kouchnirenko, A.G., Polyèdres de Newton et nombres de Milnor, Invent, math., 32(1976), 1–31.
Milne, M.C., A q-analog of the Gauss summation theorem for hypergeometric series in U(n), Ad. in Math. 72(1988), 59–131.
Mimachi, K., Connection problem in holonomic q-difference system associated with a Jackson integral of Jordan-Pochhammer type, Nagoya Math. J., 116(1989), 149–161.
Oda, T., Convex bodies and algebraic geometry — An introduction to the theory of toric varieties, Ergebnisse der Math., Springer, 1988.
Sato, M., Theory of prehomogeneous vector spaces, Algebraic part, The English version of Sato’s lecture from Shintani’s note, translated by M.Muro, to appear in Nagoya Math. J..
Slater, L.J., Generalized hypergeometric functions, Cambridge Univ. Press, 1966.
Smale, S., Algorithms for solving equations, Proc. of Inter. Congress of Math., Berkley, Cal., 1986.
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Aomoto, K., Kato, Y. (1991). A q-analogue of de Rham cohomology. In: Kashiwara, M., Miwa, T. (eds) ICM-90 Satellite Conference Proceedings. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68170-0_2
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DOI: https://doi.org/10.1007/978-4-431-68170-0_2
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