An invariance group for a linear combination of two Saalsch\"utzian ${}_4F_3(1)$ hypergeometric series

We explore a function $L(\vec{x})=L(a,b,c,d;e;f,g)$ which is a linear combination of two Saalsch\"utzian ${}_4F_3(1)$ hypergeometric series. We demonstrate a fundamental two-term relation satisfied by the $L$ function and show that the fundamental two-term 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 Saalsch\"utzian ${}_4F_3(1)$ series, and Barnes' second lemma as consequences of the fundamental two-term relation.