Relations among complementary and supplementary pairings of Saalschutzian 4F3(1) series

We investigate sums $K(\vec{x})$ and $L(\vec{x})$ of pairs of (suitably normalized) Saalschützian ${}_4F_3(1)$ hypergeometric series, and develop a theory of relations among these $K$ and $L$ functions. The function $L(\vec{x})$ has been studied extensively in the literature, and has been shown to satisfy a number of two-term and three-term relations with respect to the variable $\vec{x}$. More recent works have framed these relations in terms of Coxeter group actions on $\vec{x}$, and have developed a similar theory of two-term and three-term relations for $K(\vec{x})$. In this article, we derive "mixed" three-term relations, wherein any one of the $L$ (respectively, $K$) functions arising in the above context may be expressed as a linear combination of two of the above $K$ (respectively, $L$) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among $K$ and $L$ functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the $K$ and $L$ functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the $K$ and $L$ functions therein.

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