标题： 欧拉和克劳森恒等式的多项式扰动 显示英文标题

标题： Polynomial perturbations of Euler's and Clausen's identities

摘要： 两个超几何级数的乘积通常不是超几何的。 然而，在某些情况下，这样的乘积可以简化为一个单独的超几何级数。 这种类型的最早结果，除了显然的$(1-x)^{a}(1-x)^{b}=(1-x)^{a+b}$外，是欧拉对高斯超几何函数${}_2F_1$的变换。 另一个重要的结果是1828年提出的著名的克劳森恒等式，它将适当的${}_2F_1$函数的平方表示为一个单独的${}_3F_2$。 通过比较系数，每个乘积恒等式对应于终止级数的一种特殊的求和定理。 在过去的二十年中，通过引入相差正整数的额外参数对，欧拉变换和许多求和定理得到了扩展。 这相当于将幂级数系数乘以固定多项式在非负整数处的值。 本文的主要目标是展示通过这种多项式扰动得到的克劳森恒等式的扩展。 为此，我们首先重新考虑米勒和巴黎在2010年前后发现的欧拉变换的多项式扰动。 我们提出了他们变换的新且简化的证明，将其与多项式插值联系起来，并展示了特征多项式的各种新形式。 我们进一步引入了米勒-巴黎算子的概念，这些算子在扩展的克劳森恒等式的构造中起着重要作用。 显示英文摘要

摘要： A product of two hypergeometric series is generally not hypergeometric. However, there are a few cases when such product does reduce to a single hypergeometric series. The oldest result of this type, beyond the obvious $(1-x)^{a}(1-x)^{b}=(1-x)^{a+b}$, is Euler's transformation for the Gauss hypergeometric function ${}_2F_1$. Another important one is the celebrated Clausen's identity dated 1828 which expresses the square of a suitable ${}_2F_1$ function as a single ${}_3F_2$. By equating coefficients each product identity corresponds to a special type of summation theorem for terminating series. Over the last two decades Euler's transformations and many summation theorems have been extended by introducing additional parameter pairs differing by positive integers. This amounts to multiplication of the power series coefficients by values of a fixed polynomial at nonnegative integers. The main goal of this paper is to present an extension of Clausen's identity obtained by such polynomial perturbation. To this end, we first reconsider the polynomial perturbations of Euler's transformations found by Miller and Paris around 2010. We propose new, simplified proofs of their transformations relating them to polynomial interpolation and exhibiting various new forms of the characteristic polynomials. We further introduce the notion of the Miller-Paris operators which play a prominent role in the construction of the extended Clausen's identity.