Morris identities and the total residue for a system of type A r
Morris identities and the total residue for a system of type A r
复制标题
A r 型系统的 Morris 恒等式和总残基
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
M. Vergne
中科院分区：
文献类型：
--
作者：
Velleda Baldoni;M. Vergne
The purpose of this paper is to find explicit formulae for the total residue of some interesting rational functions with poles on hyperplanes determined by roots of type A r = {(e i −e j )|1 ≤ i, j ≤ (r+1), i ≠ j}. As pointed out by Zeilberger [Z], these calculations are mere reformulations of Morris identities [M], where the total residue function replaces here the iterated constant term. The proof we give of these identities follows closely (as suggested in [Z] Aomoto’s computation [Aom] of generalized Selberg integrals. Recall that Selberg [Se] proved that the following integral:
$$ {S_{r}}left( {{k_{1}},{k_{2}},{k_{3}}}
ight) = int_{{{{[0,1]}^{r}}}} {prodlimits_{{i = 1}}^{r} {x_{i}^{{{k_{1}}}}{{(1 - {x_{i}})}^{{{k_{2}}}}}} } prodlimits_{{1 le i < j le r}} {|({x_{i}} - {x_{j}}){|^{{{k_{3}}}}}dx} $$
is a product of Г functions. In this setting, k 1 , k 2 , k 3 are nonnegative integers. Here we will be interested in the Fourier transform of the function
$$ phi ({k_{1}},{k_{2}},{k_{3}})({x_{0}},x) = frac{1}{{prod
olimits_{{i = 1}}^{r} {x_{i}^{{{k_{1}}}}{{({x_{0}} - {x_{i}})}^{{{k_{2}}}}}{{prod
olimits_{{1 le i < j le r}} {({x_{i}} - {x_{j}})} }^{{{k_{3}}}}}} }}, $$
more particularly on the value
$${{s}_{r}}({{k}_{1}},{{k}_{2}},{{k}_{3}})(xi ) = int_{{{{mathbb{R}}^{{r + 1}}}}} {{{e}^{{i{{x}_{0}}xi }}}phi ({{k}_{1}},{{k}_{2}},{{k}_{3}})({{x}_{0}},x)d{{x}_{0}}dx}$$
where ϕ (k 1, k 2, k 3) (x 0, x) is interpreted as a boundary value of an holomorphic function. As shown by Jeffrey–Kirwan [JK], the value of the function s r (k 1, k 2, k 3) (ξ) is easily deduced from the knowledge of the total residue of the integrand. This reduces the problem to purely algebraic consideration: “integration” means that we will explicitly compute the function ϕ (k 1, k 2, k 3) (x 0, x) modulo derivatives in x 1, x 2..., x r according to the decomposition appearing in Equation 1.1.
The purpose of this paper is to find explicit formulae for the total residue of some interesting rational functions with poles on hyperplanes determined by roots of type A r = {(e i −e j )|1 ≤ i, j ≤ (r+1), i ≠ j}. As pointed out by Zeilberger [Z], these calculations are mere reformulations of Morris identities [M], where the total residue function replaces here the iterated constant term. The proof we give of these identities follows closely (as suggested in [Z] Aomoto’s computation [Aom] of generalized Selberg integrals. Recall that Selberg [Se] proved that the following integral:
$$ {S_{r}}left( {{k_{1}},{k_{2}},{k_{3}}}
ight) = int_{{{{[0,1]}^{r}}}} {prodlimits_{{i = 1}}^{r} {x_{i}^{{{k_{1}}}}{{(1 - {x_{i}})}^{{{k_{2}}}}}} } prodlimits_{{1 le i < j le r}} {|({x_{i}} - {x_{j}}){|^{{{k_{3}}}}}dx} $$
is a product of Г functions. In this setting, k 1 , k 2 , k 3 are nonnegative integers. Here we will be interested in the Fourier transform of the function
$$ phi ({k_{1}},{k_{2}},{k_{3}})({x_{0}},x) = frac{1}{{prod
olimits_{{i = 1}}^{r} {x_{i}^{{{k_{1}}}}{{({x_{0}} - {x_{i}})}^{{{k_{2}}}}}{{prod
olimits_{{1 le i < j le r}} {({x_{i}} - {x_{j}})} }^{{{k_{3}}}}}} }}, $$
more particularly on the value
$${{s}_{r}}({{k}_{1}},{{k}_{2}},{{k}_{3}})(xi ) = int_{{{{mathbb{R}}^{{r + 1}}}}} {{{e}^{{i{{x}_{0}}xi }}}phi ({{k}_{1}},{{k}_{2}},{{k}_{3}})({{x}_{0}},x)d{{x}_{0}}dx}$$
where ϕ (k 1, k 2, k 3) (x 0, x) is interpreted as a boundary value of an holomorphic function. As shown by Jeffrey–Kirwan [JK], the value of the function s r (k 1, k 2, k 3) (ξ) is easily deduced from the knowledge of the total residue of the integrand. This reduces the problem to purely algebraic consideration: “integration” means that we will explicitly compute the function ϕ (k 1, k 2, k 3) (x 0, x) modulo derivatives in x 1, x 2..., x r according to the decomposition appearing in Equation 1.1.