On a theta product formula for the symmetric \(A\)-type connection function
A formula which gives a product expression for a sum of theta rational functions is obtained. Such a sum appears in the connection formulae among symmetric \(A\)-type Jackson integrals. Let \(\{U_ \sigma (x)\}_{\sigma \in {\mathcal G}_ n}\) be the set of theta rational functions defined on the \(n\)-dimensional algebraic torus \((C^*)^ n\), where \({\mathcal G}_ n\) is the symmetric group of \(n\)-th degree. For the theta rational function \(\varphi (x)\), \(x = (x_ 1, \dots, x_ n) \in (C^*)^ n\), defined by the formula \[ \varphi (x) = \prod^ n_{j = 1} x_ j^{\alpha_ j} {\theta (2^{ \alpha_ j + \cdots + \alpha_ n + \gamma + 1} x_ j/x_{j - 1}) \over \theta (2^{\gamma + 1} x_ j/x_{j - 1})} \] (with \(x_ 0 = q^ \gamma)\), the following generalized alternating sum with the weight \(\{U_ \sigma^{-1} (x)\}_{\sigma \in {\mathcal G}_ n}\) is introduced: \(\widetilde \varphi (x) = \sum_{ \sigma \in{\mathcal G}_ n} \sigma \varphi (x) \cdot \text{sign} (\sigma) \cdot U_ \sigma^{-1} (x)\). Then, it is shown that \(\widetilde \varphi (x)\) can be expressed as a product of theta monomials. This result has been proved by the author in a previous paper for the particular cases \(n = 2\) and \(n = 3\). The formula giving \(\widetilde \varphi (x)\) can be regarded as an elliptic version of the one concerning Hall-Littlewood polynomials.
scientific article; zbMATH DE number 755435