Coxeter group actions and limits of hypergeometric series
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Coxeter group actions and limits of hypergeometric series R. M. Green1 · Ilia D. Mishev1 · Eric Stade1 Received: 31 December 2018 / Accepted: 17 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised 9 F8 hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group W (E 7 ) on this set. We show that, by removing from W (E 7 ) a particular generator, we obtain a subgroup that is isomorphic to W (D6 ), and that acts intransitively on S, partitioning it into three orbits, of sizes 32, 12, and 12, respectively. Taking certain limits of the M functions in the first orbit yields a set of 32 J functions, each of which is a sum of two Saalschützian 4 F3 hypergeometric series of unit argument. The original action of W (D6 ) on the M functions in this orbit is then seen to correspond to a known action of this group on this set of J functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 L functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one M function from each size-12 orbit. Finally, each known three-term relation among the J and L functions is seen to be obtainable as a limit of a known three-term relation among the M functions. Keywords Hypergeometric series · Coxeter groups · Functional equations Mathematics Subject Classification 33C20 · 20F55
B 1
Eric Stade [email protected] University of Colorado, Boulder, CO, USA
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R. M. Green et al.
1 Introduction The hypergeometric series F(a, b; c; z)
ab z 1!c a(a + 1)b(b + 1) 2 z + 2!c(c + 1) a(a + 1)(a + 2)b(b + 1)(b + 2) 3 z + ··· + 3!c(c + 1)(c + 2) =1+
(1.1)
was introduced and studied by Gauss [8] in 1821. Gauss derived many properties of this series, and demonstrated its relationship to a wide array of elementary and special functions. The “Gauss function,” as (1.1) is now known, quickly became ubiquitous in mathematics and the physical sciences. The theory of generalized hypergeometric series—series like (1.1), but with arbitrary numbers of numerator and denominator parameters—began to take form in the latter part of the 1800’s. (The series (1.1) has two numerator parameters, a and b, and one denominator parameter, c.) From this period through the early part of the 1900’s, properties of—and, especially, relations among—these generalized series were studied extensively. (See [1–3,23,26,27], and [28], to name just a few.) More recently, generalized hypergeometric series—particularly those with unit argument, meaning z = 1—have figured prominently in various other contexts. For
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