Some basic bilateral sums and integrals.
Some basic bilateral sums and integrals.
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DOI：
10.2140/pjm.1995.170.497
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发表时间：
1993-11
影响因子：
0.6
通讯作者：
M. Ismail;Mizan Rahman
中科院分区：
文献类型：
--
作者：
M. Ismail;Mizan Rahman
通过将真实的直线分割成单位长度的区间，形式为$\IntF（q^x）\，dx，\; 0<q<1$的双无穷积分可以清楚地表示为$\Integ \Sum F（q^{x+n}）\，dx$，只要$F$满足适当的条件。利用这个简单的思想证明了Ramanujan\ph{1}{1}和的积分类似，并给出了Askey和Roy的推广的一个新的证明，同时还以一种简单的方式找到了良平衡\ph{2}{2}和以及非常良平衡\ph{6}{6}和的积分类似.一个扩展到一个非常良好的平衡和平衡\ph{8}{8}系列也给出了。给出了Ismail和Masson最近提出的一个q-β积分的直接证明。
By splitting the real line into intervals of unit length a doubly infinite integral of the form $\Int F(q^x)\,dx,\; 0<q<1$, can clearly be expressed as $\Integ \Sum F(q^{x+n})\,dx$, provided $F$ satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his \ph{1}{1} sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised \ph{2}{2} sum as well as the very-well-poised \ph{6}{6} sum are also found in a straightforward manner. An extension to a very-well-poised and balanced \ph{8}{8} series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.