Board index Computation of Integrals Multiple integrals and parametric series

Multiple integrals and parametric series

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Multiple integrals and parametric series

Sun Aug 21, 2016 1:11 am

Posts: 105
How to prove the following identity
$\sum\limits_{l = 0}^\infty {\frac{1}{{{{\left( {l + \alpha } \right)}^n}\left( {l + \beta } \right)}}} = \int\limits_0^1 {\frac{{{{\left( {1 - {t_0}} \right)}^{\beta - 1}}}}{{t_0^\alpha }}d{t_0}} \int\limits_0^{{t_0}} {\frac{{d{t_1}}}{{{t_1}}}} \cdots \int\limits_0^{{t_{n - 2}}} {\frac{{d{t_{n - 1}}}}{{{t_{n - 1}}}}} \int\limits_0^{{t_{n - 1}}} {\frac{{t_n^{\alpha - 1}}}{{{{\left( {1 - {t_n}} \right)}^\beta }}}} d{t_n},$
where $\alpha ,\beta > - 1.$