$q$-analogue of de Rham cohomology associated with Jackson integrals, II
$q$-analogue of de Rham cohomology associated with Jackson integrals, II
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$q$-与 Jackson 积分相关的 de Rham 上同调的类似物，II
DOI：
10.3792/pjaa.66.240
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发表时间：
1990
影响因子：
0.9
通讯作者：
K. Aomoto
中科院分区：
文献类型：
--
作者：
K. Aomoto
在这篇文章中，我们想通过经典的巴恩斯表示给出涉及基本超几何函数的杰克逊积分的一个新公式。我们定义了一个可以用Sato的b函数的q版来表示的de Rham上同调的q-类似，并推导了相应的完整q-差分系统。它的多重性的评价将以一些不同的渐近给出。1. b函数的结构。我们取椭圆模q- e， Imr0。设X是一个n维整数晶格-Z。令X= X(R)C*，被q扭曲的n维代数环面，设z，，…， Z是X的一组基，使得n任意zxe可以唯一地写成=__，Z,, ezz。我们可以用n个C*的直积来标识X同构于X(R) （C/ (2i log q)）。将Z与元素t=(1,)识别，可得到包含物XcX。， 1, q, 1。， 1) e （C*）。我们用Q表示移位算子Qf(t)=f(。T)由位移T -引起。t表示x上的函数，我们把Qz——Q?…Q。我们考虑q-差分方程（1.1）QzO(t)=-bz(t)CP(t)， zex和tex，或X上的一组不等于零的有理函数{b(t)}e。{b(t)}ez满足相容条件（1.2）b /,(t)—b(t) Qb,(t)，使得{b(t)}zex定义了X上的l-环，其值在R(X)上是由X上的非零有理函数组成的乘法abelin群。我们用R(X)表示X上的0个有理函数域。{b(t)}zex是共边界当且仅当b(t)=Q(t)/o(t)对于e R(X)。我们用H（X， R(X)）表示相应的l上同。我们将(x)= 1-[%0 （1- xq9）和(x)=(x)/(xq)代入n e z，则以下重要结果成立。命题。任意的协环{bz(t)}zex模一个协边界可以用式（1.1）表示，其中表示X上的q乘函数，表示为
In this note we want to 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 formulated by means of q-version of Sato’s b-functions and derive associated holonomic q-difference system. The evaluation of its multiplicity will be given as a number of different asymptotics. 1. Structure of b-functions. We take the elliptic modulus q--e, Imr0. Let X be an n dimensional integer lattice -Z. We put X= X(R)C*, the n dimensional algebraic torus twisted by q. Let z, , ..., Z be basis of X such that n arbitrary Z e X can be uniquely written by =__,z, , e Z. We may identify X isomorphic to X(R) (C/ (2i log q)) with the direct product of n pieces o C*. The inclusion XcX can be obtained by identifying Z with the element t=(1, ., 1, q, 1, ., 1) e (C*). We denote by Q the shift operator Qf(t)=f(. t) induced by the displacement t-.t for a unction f on X. We put Qz--Q?...Q. We consider the q-difference equations (1.1) QzO(t)=-bz(t)CP(t), z e X and t e X, or a set of rational functions {b(t)}e, on X, which are not identically zero. {b(t)}ez satisfies the compatibility condition (1.2) b / ,(t)--b(t) Qb,(t), so that {b(t)}zex defines l-cocycle on X with vlues in R(X) the multiplicative abelin group consisting of non-zero rational functions on X. We denote by R(X) the field o rational unctions on X. {b(t)}zex is a coboundary if and only if b(t)=Q(t)/o(t) for e R(X). We write the corresponding l-cohomology by H(X, R(X)). We put (x)= 1-[%0 (1--xq9 and (x)=(x)/(xq) for n e Z. Then the following important result holds. Proposition. An arbitrary cocycle {bz(t)}zex modulo a coboundary can be expressed by (1.1), where denotes a q-multiplicative function on X written by