Let $p$ and $q$ be two positive integers and let $\beta \neq 1/2$ be a real number. Then let $B > A > 0$. With the help of Mathematica, ie by doing elementary integrations and by consecutively simplifing the results I have established a following identity: \begin{eqnarray} &&\int\limits_{A \le \xi_0 \le \xi_1 \le \cdots \le \xi_{p-1} \le B} \prod\limits_{j=0}^p (\xi_{j-1}-\xi_j)^q \cdot \prod\limits_{j=0}^{p-1} \frac{d \xi_j}{\xi_j^{2\beta}} =\sum\limits_{m=0}^p \sum\limits_{l=m(q+1-2 \beta)}^{q+m(q+1-2 \beta)} \\&&\binom{q}{l-m(q+1-2\beta)} A^{q+p(q+1-2\beta)-l} B^l (-1)^{l+m(2\beta-1)} \frac{(-1)^{(p+m)q}}{(q+1)^p \prod\limits_{\eta=1}^p \binom{l+(2\beta-q-1)(\eta-1+1_{\eta>m})}{q+1}} \end{eqnarray} subject to $\xi_{-1}=A$ and $\xi_p=B$. Note that the right hand side consists of $(p+1)(q+1)$ terms. This identity is a generalization of similar identity given in Multivariable integral over a simplex . The result can be readily used to find closed form solutions to linear ordinary differential equations with time varying coefficients, as I will show in the future. Now, given that the result has a very neat structure the question is can it be derived using some other method rather than tedious elementary integration and simplification. In particular does the result have a geometric interpretation?
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