2006/02/28 by Michael Schlosser, Michael J. Schlosser · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Bayesian Methods and Mixture Models #Binomial coefficient #Combinatorics #Computer science #Degenerate energy levels #Discrete mathematics #Elliptic function #Enumeration #Half-integer #Integer lattice #Lattice (music) #Mathematical analysis #Mathematics #Physics #Planar #Pure mathematics #Quantum mechanics #Unit circle #math.CA #math.CO #msc:05A15 #msc:05A17 #msc:05A19 #msc:05E10 #msc:11B65 #msc:33D15 #msc:33E20
paper · pdf · doi:10.1016/j.jcta.2006.07.002
published as J. Combin. Theory Ser. A 114 (3) (2007), 505-521 · minor changes, 17 pages, to appear in JCTA
arxiv created 2006/07/15 · openalex publication_date 2006/08/24 · arxiv updated 2019/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's 10-V-9 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's 8-phi-7 and Dougall's 7-F-6 summation. By considering nonintersecting lattice paths we are led to a multivariate extension of the 10-V-9 summation which turns out to be a special case of an identity originally conjectured by Warnaar, later proved by Rosengren. We conclude with discussing some future perspectives.