A q-integral of Selberg and Askey
A q-integral of Selberg and Askey
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Selberg 和 Askey 的 q 积分
DOI：
10.1137/0519111
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发表时间：
1988
影响因子：
2
通讯作者：
L. Habsieger
中科院分区：
文献类型：
--
作者：
L. Habsieger
We prove a conjecture by R. Askey (“Some basic hypergeometric extensions of integrals of Selberg and Andrews,” SIAM J. Math. Anal., 11(1980), pp. 938–951) on a basic extension of Selberg’s integral: \[ \int_0^1 \cdots \int_0^1 {\mathop \prod \limits_{1 \leqq i < j \leqq n} } \left| {t_i - t_j } \right|^{2z} \mathop \prod \limits_{i = 1}^n t_i^{x - 1} \left(1 - t_i \right)^{y - 1} dt_1 \cdots dt_n . \] We deduce from this a conjecture due to Morris about the constant term in the expansion of \[ \mathop \prod \limits_{j = 1}^l \left( {{{x_0 } / {x_j }}} \right)_a \left( {{{qx_j } / {x_0 }}} \right)_b \mathop \prod \limits_{1 \leqq i < j \leqq l} \left( {{{x_i } / {x_j }}} \right)_c \left( {{{qx_j } / {x_i }}} \right)_c ,\] where $(x)_k = (1 - x)(1 - qx) \cdots (1 - q^k x)$. In the appendix there can be found a proof of another conjecture by Askey related to the Dyson q-conjecture.
