On a k-Fold Beta Integral Formula

Abstract

In this paper, we investigate some necessary and sufficient conditions which ensure validity of a k-fold Beta integral formula for any $$x^{i}\in \mathbb {R}^n$$ and some nonzero real numbers $$d_i$$ with $$i=1,2,\ldots ,k$$ . We establish that Eq. (0.1) holds if and only if and This yields a complete answer to the question raised by Grafakos and Morpurgo in [4]. In addition, it turns out that formula (0.1) does not hold if $$k\ge 4$$ . For those $$k, d_1, d_2, \ldots , d_k$$ not satisfying (0.2) or (0.3), we prove that the real-valued integral $$\displaystyle {\int _{\mathbb {R}^n}\prod \limits ^k_{i=1}|x^{i}-t|^{-d_i}dt}$$ can be represented as a function of distances of consecutive differences of the sequence $$x^1, x^2, \ldots , x^k$$ .