Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of $L$-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove that, in certain cases, these unitary averages factor as polynomials into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the “swap” terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from random matrix theory.

数论学家对经典紧群上平均的平移特征多项式的乘积以及此类乘积的比率非常感兴趣，因为他们对与紧群具有相同对称类型的族中的 $L$ 函数的相似平均值进行建模团体。我们使用 Toeplitz 和 Toeplitz plus Hankel 算子以及 Borodin-Okounkov-Case-Geronimo 和 Basor-Ehrhardt 的恒等式来证明，在某些情况下，这些酉平均值可以作为多项式分解为辛群和正交群的平均值。在这些恒等式的基础上，我们提出了这些平均值的精确公式的新证明，其中作为数论平均值特征的“交换”项来自适当 Hankel 算子行列式的 Fredholm 展开。这是平移特征多项式乘积比平均值公式的第四种不同证明；其他证明基于超对称性；对称函数理论和随机矩阵理论的正交多项式方法。