An invariance group for a linear combination of two Saalschützian ${}_4F_3(1)$ hypergeometric series
Abstract
We explore a function $L(\vec{x})=L(a,b,c,d;e;f,g)$ which is a linear combination of two Saalschützian ${}_4F_3(1)$ 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 Saalschützian ${}_4F_3(1)$ series, and Barnes' second lemma as consequences of the fundamental two-term relation.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2009
- DOI:
- 10.48550/arXiv.0910.0093
- arXiv:
- arXiv:0910.0093
- Bibcode:
- 2009arXiv0910.0093M
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- 33C20
- E-Print:
- 15 pages