赞美欧拉的一个初等恒等式
In Praise of an Elementary Identity of Euler
摘要：
我们调查的应用程序的一个基本身份所使用的欧拉在他的证明之一的五边形数定理。使用一个适当的重新制定的版本，这个身份，我们称之为欧拉的望远镜引理，我们给所有的关键求和定理的替代证明终止超几何级数和基本超几何级数，包括终止二项式定理，楚-范德蒙和，普法夫-Saalch\”utz和，和他们的$q$ -类似物。我们还证明了杰克逊的$q$ -模拟杜格尔的总和，一个终止的，平衡的，非常平衡的$_8ϕ_7$总和。我们的证明在概念上与WZ方法相同，但不使用计算机。本文综述了Macdonald给出的广义超几何级数的恒等式，并证明了组合中出现的斐波那契数、多项式和Pell数的类似物$q$的几个恒等式.其中一些身份似乎是新的。
英文摘要：
We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8ϕ_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.
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