Stanley's Lemma and Multiple Theta Functions

2013/12/08 by William Y. C. Chen, Chen, William Y. C., Lisa H. Sun +1
Mathematics · #05E45 #14K25 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #math.CA #math.CO #msc:05E45 #msc:14K25

paper · pdf · doi:10.48550/arxiv.1312.2172

33 pages; to appear in SIAM J. Discrete Math

arxiv created 2017/07/09 · arxiv updated 2017/07/11

Abstract

We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions [(-1)^δa₁^α₁a₂^α₂⋯ a_r^α_rq^s; q^t]_∞, where α_i are integers, δ=0 or 1, s∈ ℚ, t∈ ℚ⁺, and the exponent vectors (α₁,α₂,…,α_r) are linearly independent over ℚ. For an identity on such multiple theta functions, we provide an algorithmic approach for computing a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's Lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a finite number of its coefficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity.

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