Coxeter Symmetry in Hypergeometric Functions and Elliptic Integral Moments

Source abstract

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.