Coxeter Symmetry in Hypergeometric Functions and Elliptic Integral Moments
Abstract
We develop a hypergeometric framework for elliptic integral moments organized around Mishev's completed Saalschützian L-function and its W(D5) symmetry. The Bailey--Mishev very-well-poised 7F6(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 s4,0. 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(D5) symmetry of generalized hypergeometric functions.
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BibTeX
@misc{bao2026coxeter,
title = {{Coxeter Symmetry in Hypergeometric Functions and Elliptic Integral Moments}},
author = {Dianbin Bao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34016},
eprint = {2609.34016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34016},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coxeter Symmetry in Hypergeometric Functions and Elliptic Integral Moments
AU - Dianbin Bao
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.34016
UR - https://arxiv.org/abs/2609.34016
AB - 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.
ER -