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Coxeter group actions on Saalschützian ${}_4F_3(1)$ series and very-well-poised ${}_7F_6(1)$ series

In this paper we consider a function $L(\vec{x})=L(a,b,c,d;e;f,g)$, which can be written as a linear combination of two Saalschützian ${}_4F_3(1)$ hypergeometric series or as a very-well-poised ${}_7F_6(1)$ hypergeometric series. We explore two-term and three-term relations satisfied by the $L$ function and put them in the framework of group theory. We prove a fundamental two-term relation satisfied by the $L$ function and show that this relation implies that the Coxeter group $W(D_5)$, which has 1920 elements, is an invariance group for $L(\vec{x})$. The invariance relations for $L(\vec{x})$ are classified into six types based on a double coset decomposition of the invariance group. The fundamental two-term relation is shown to generalize classical results about hypergeometric series. We derive Thomae's identity for ${}_3F_2(1)$ series, Bailey's identity for terminating Saalschützian ${}_4F_3(1)$ series, and Barnes' second lemma as consequences. We further explore three-term relations satisfied by $L(a,b,c,d;e;f,g)$. The group that governs the three-term relations is shown to be isomorphic to the Coxeter group $W(D_6)$, which has 23040 elements. Based on the right cosets of $W(D_5)$ in $W(D_6)$, we demonstrate the existence of 220 three-term relations satisfied by the $L$ function that fall into two families according to the notion of $L$-coherence.