梅林空间反射、椭圆Γ函数的模性及其他
Mellin space reflections, modularity of elliptic Gamma functions and beyond
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中文总结 AI 辅助
该研究从梅林空间视角将模变换组织为梅林核反射恒等式，扩展至椭圆Γ函数，得到关联SL(3,ℤ)模公式中三个椭圆Γ函数的三重线性反射恒等式，为研究模特殊函数提供了新原则。
中文摘要 AI 辅助
我们从梅林空间的视角研究模变换公式。基本观察是，赫维茨ζ函数与多重对数之间的函数关系可用于将模变换组织为梅林核的反射恒等式，而伴随的多项式异常则源于轮廓变形。我们首先以q-θ函数为例说明这一机制，随后将其扩展至椭圆Γ函数。在后一情形中，对两个椭圆方向分别进行梅林变换，会得到一个三重线性反射恒等式，它关联SL(3,ℤ)模公式中出现的三个椭圆Γ函数；同时，相关的轮廓变形会重现三次伯努利多项式。进一步的反射公式会产生一类带权q-Pochhammer型函数，其模变换与q-θ函数类似，还会产生一个转置的三重线性反射恒等式，与SL(3,ℤ)变换背后的梅林空间结构类似。我们的结果表明，梅林空间反射恒等式为构造和研究标准多重椭圆Γ函数层级之外的模特殊函数提供了一种有用的组织原则。
英文摘要
We study modular transformation formulas from the viewpoint of Mellin space. The basic observation is that the functional relation between the Hurwitz zeta function and the polylogarithm can be used to organize modular transformations as reflection identities of Mellin kernels, while the accompanying polynomial anomalies arise from contour deformations. We first illustrate this mechanism for the $q$-$θ$ function and then extend it to the elliptic Gamma function. In the latter case, independently Mellin transforming the two elliptic directions leads to a trilinear reflection identity relating the three elliptic Gamma functions appearing in the SL$(3,\mathbb{Z})$ modular formula, while the associated contour deformation reproduces the cubic Bernoulli polynomial. Further reflection formulas lead to a weighted $q$-Pochhammer type function with a modular transformation analogous to that of the $q$-$θ$ function, as well as a transposed trilinear reflection identity analogous to the Mellin space structure underlying the SL$(3,\mathbb{Z})$ transformation. Our results suggest that Mellin space reflection identities provide a useful organizing principle for constructing and studying modular special functions beyond the standard multiple elliptic Gamma hierarchy.
