Stanley's Lemma and Multiple Theta Function Identities
William Y. C. Chen and Lisa H. Sun
Abstract: We present a vector space approach to proving identities on multiple theta functions by verifying a finite number of simpler relations which are often verifiable by using Jacobi's triple product identity. Consider multiple theta function identities of the form ∑^{m}_{k=1} c_{k}θ_{k}(a_{1}, a_{2},..., a_{r}) = 0, where r, m ≥ 2, θ_{k} = ∏^{nk}_{i=1} f_{k,i}(a_{1}, a_{2},..., a_{r}), 1 < n_{k} ≤ r and each f_{k,i} is of form (p_{k,i}, q^{βk,i}/p_{k,i}; q^{βk,i})1 with β_{k,i} being a positive integer and p_{k,i} being a monomial in a_{1}, a_{2},... a_{r} and q. For such an identity, θ_{1}, θ_{2},..., θ_{m} satisfy the same set of linearly independent contiguous relations. Let W be the set of exponent vectors of (a_{1}, a_{2},..., a_{r}) in the contiguous relations. We consider the case when the exponent vectors of (a_{1}, a_{2},..., a_{r}) in p_{k,1}, p_{k,2},..., p_{k,nk} are linearly independent for any k. Let V_{C} denote the vector space spanned by multiple theta functions which satisfy the contiguous relations associated with the vectors in W. Using Stanley's lemma on the fundamental parallelepiped, we find an upper bound of the dimension of V_{C}. This implies that a multiple theta function identity may be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, such as Bailey's generalization of the quintuple product identity, the extended Riemann identity and Riemann's addition formula. AMS Classification: 05E45, 14K25 Keywords: theta function, multiple theta function, contiguous relation, Jacobi's triple product identity, addition formula Download: pdf |