发表机构
Department of Mathematics, Alvernia University（阿尔弗尼亚大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立超几何框架，利用$W(D_5)$对称性统一推导椭圆积分矩、调和和及特殊值恒等式，揭示其共同对称性来源。
AI 中文摘要
我们发展了一个围绕Mishev的完整Saalschützian $L$-函数及其$W(D_5)$对称性组织的椭圆积分矩的超几何框架。Bailey--Mishev非常良态${}_7F_6(1)$表示将互补的椭圆积分矩和调和超几何和置于共同的参数空间中。我们首先给出基于碰撞的超几何推导，涉及$K'^2$、$E'K'$和$E'^2$的三个互补模矩族。$E'^2$恒等式也独立地从镜像椭圆积分微分系统的Barnes-相邻关系中恢复。然后我们推导四贝塞尔矩$s_{4,0}$的调和级数表示的超几何评估。在对称参数点，Coxeter对称性迫使选定的Taylor系数沿对称适应路径消失，产生涉及Riemann zeta函数值和同一贝塞尔周期的调和及特殊值恒等式族。所得图像表明这些椭圆矩、调和和与特殊值恒等式源于广义超几何函数的共同$W(D_5)$对称性。
英文摘要
We develop a hypergeometric framework for elliptic integral moments organized around Mishev's completed Saalschützian $L$-function and its $W(D_5)$ symmetry. The Bailey--Mishev very-well-poised ${}_7F_6(1)$ representation places complementary elliptic integral moments and harmonic hypergeometric sums in a common parameter space. We first give collision-based hypergeometric derivations of the three complementary-modulus moment families involving $K'^2$, $E'K'$, and $E'^2$. The $E'^2$ identity is also recovered independently from a Barnes-contiguous relation mirroring the differential system of the elliptic integrals. We then derive a hypergeometric evaluation of the harmonic-series representation of the four-Bessel moment $s_{4,0}$. At symmetric parameter points, Coxeter symmetry forces selected Taylor coefficients to vanish along symmetry-adapted paths, producing families of harmonic and special-value identities involving values of the Riemann zeta function and the same Bessel period. The resulting picture shows that these elliptic-moment, harmonic-sum, and special-value identities arise from a common $W(D_5)$ symmetry of generalized hypergeometric functions.
Comments41 pages
