Stanley's Lemma and Multiple Theta Functions

Abstract

We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions $[(-1{)}^{\delta }{a}_1^{{\alpha }_1}{a}_2^{{\alpha }_2}\cdots a_r^{{\alpha }_r}{q}^s;q^t{]}_{\infty }$, where ${\alpha }_i$ are integers, $\delta =0$ or $1$, $s\in \mathbb{Q}$, $t\in {\mathbb{Q}}^{+}$, and the exponent vectors $({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_r)$ are linearly independent over $\mathbb{Q}$. 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.

33 pages; to appear in SIAM J. Discrete Math