The Cauchy operator for basic hypergeometric series
The Cauchy operator for basic hypergeometric series
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DOI：
10.1016/j.aam.2007.08.001
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发表时间：
2007-05
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影响因子：
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通讯作者：
Vincent Y. B. Chen;Nancy S. S. Gu-Nancy-S.-S.-Gu-145451531
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文献类型：
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作者：
Vincent Y. B. Chen;Nancy S. S. Gu-Nancy-S.-S.-Gu-145451531
我们引入了基本超几何级数的柯西增广算子。利用算子恒等式中某些参数的对称性，可以很容易地得到海涅的12变换公式和西尔斯的23变换公式. Cauchy算子包含两个参数，它可以被认为是算子T（bDq）的推广。利用这个算子，我们得到了Askey-Wilson积分，Askey-Roy积分，Sears的两项求和公式以及巴恩斯引理的q-类似.最后，我们发现Cauchy算子也适用于研究二元Rogers-Szegö多项式，或连续的大q-Hermite多项式.
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ϕ12 transformation formula and Sears'ϕ23 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bDq). Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.