2010/08/05 by Ilia D. Mishev, Mishev, Ilia D.
Mathematics · #20F55 #33C20 #33C60 #33C80 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CA #math.CO #math.GR #msc:20F55 #msc:33C20 #msc:33C60 #msc:33C80
paper · pdf · doi:10.48550/arxiv.1008.1011
20 pages
arxiv created 2010/08/05 · arxiv updated 2010/08/06
In this paper we consider a function L(x)=L(a,b,c,d;e;f,g), which can be written as a linear combination of two Saalschützian 4F3(1) hypergeometric series or as a very-well-poised 7F6(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(D5), which has 1920 elements, is an invariance group for L(x). The invariance relations for L(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 3F2(1) series, Bailey's identity for terminating Saalschützian 4F3(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(D6), which has 23040 elements. Based on the right cosets of W(D5) in W(D6), 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.