Measures of Dirichlet type on regular polygons and their moments (Q1191002)

Consider \[ P_ k=\prod_{j=0}^{k-1} (1-x_ j)^{-\alpha}=(1-x_{k- 1})^{-k\alpha} \prod_{j=0}^{k-2} \left[1-{{x_ j-x_{k-1}} \over {1-x_{k-1}}}\right]^{-\alpha}. \] The latter product, being a special \(F_ D^{(k-1)}\), is expressible in terms of the customary \((k-1)\)- dimensional Dirichlet integral. This in turn is recast as \[ \int_{E_ k} \left[1-\sum_{k=0}^{k-1} x_ j t_ j\right]^{-k\alpha} d\mu_ \alpha, \] where \(E_ k\) is the simplex described by barycentric coordinates \(t_ 0,\dots,t_{k-1}\), while \(d\mu_ \alpha\) is a suitable measure. Now, taking \(x_ j\) equal to \(a\omega^ j + b\omega^{-j}\), where \(\omega=\exp{2\pi i\over k}\) and \(\max\{| a|,| b|\}<{1\over 2}\), the author succeeds in transforming \(P_ k\) into a triple hypergeometric series \(W\) of order \(k\) in the variables \((a^ k,b^ k,ab)\). Moreover, the integral over \(E_ k\) is transformed into an integral over the unit \(k\)-gon \(X_ k\subset\mathbb{C}\) with the same measure; and the main result then takes the form \[ \int_{X_ k}(1-(az+b\bar z))^{-k\alpha} d\mu_ \alpha =W. \] By expansion of the integrand, the author then derives a finite sum expressing the nonzero moments \(\int_{X_ k} z^{ks+t_{\bar z}t} d\mu_ \alpha\), where \(s,t\in\mathbb{Z}_ +\). Finally, the case of even \(k\) is given further consideration along similar lines.