标题： 孙-潘型循环消失恒等式：分析与模观点 显示英文标题

标题： Cyclic Vanishing Identities of Sun-Pan Type: Analytic and Modular Perspectives

摘要： 我们重新审视Sun--Pan型的伯努利多项式的循环恒等式及其$q$-类似物。 从分析的角度来看，我们提出了强制循环消失恒等式的最小Appell公理，自然扩展到$q$-Appell序列和解析伯努利函数。 从模形式的角度来看，我们证明了相同的关系作为与Eisenstein级数相关的周期多项式恒等式，反映了模群的对称性$(ST)^3=-I$。 这两种互补的视角将Sun--Pan循环恒等式置于数论、特殊函数和模形式的交叉点，并暗示了与多对数函数、$L$-值和混合Tate动机的进一步联系。 显示英文摘要

摘要： We revisit the cyclic identities of Sun--Pan type for Bernoulli polynomials and their $q$-analogues. From the analytic side, we formulate minimal Appell axioms that force cyclic vanishing identities, extending naturally to $q$-Appell sequences and analytic Bernoulli functions. From the modular side, we show that the same relations arise as period polynomial identities associated with Eisenstein series, reflecting the symmetry $(ST)^3=-I$ of the modular group. These two complementary perspectives place the Sun--Pan cyclic identities at the crossroads of number theory, special functions, and modular forms, and suggest further connections to polylogarithms, $L$-values, and mixed Tate motives.