Rook theory has been investigated by many people since its introduction byKaplansky and Riordan in 1946. Goldman, Joichi and White in 1975 showed thatthe sum over $k$ of the product of the $(n-k)$-th rook numbers multiplied bythe $k$-th falling factorial polynomials factorize into a product. In thesequel, different types of generalizations and analogues of this productformula have been derived by various authors. In 2008, Miceli and Remmelconstructed a rook theory model involving augmented rook boards in which theyshowed the validity of a general product formula which can be specialized toall other product formulas that so far have appeared in the literature on rooktheory. In this work, we construct an elliptic extension of the $q$-analogue ofMiceli and Remmel's result. Special cases yield elliptic extensions of variousknown rook theory models.
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机译：自1946年Kaplansky和Riordan引入Rook理论以来，许多人对其进行了研究。Goldman，Joichi和White于1975年提出，第（nk）$子菜数字乘积$ k $乘以$ k $。降阶阶乘多项式分解为乘积。在这些问题中，不同作者得出了该产品公式的不同类型的概括和类似物。在2008年，Miceli和Remmel构建了一个包含增强型白嘴鸦板的白嘴鸦理论模型，在该模型中，他们证明了通用产品公式的有效性，该公式可以专门用于到目前为止在白嘴鸦理论上出现的所有其他产品公式。在这项工作中，我们构造了Miceli和Remmel结果的$ q $模拟的椭圆扩展。特殊情况会产生各种已知的车鸦理论模型的椭圆扩展。
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