Background: I'm trying to compute some harmonic sums using Fourier Legendre expansion. For instance, the following expansion $$\frac{\log (1-x)}{x}=\sum _{n=0}^{\infty } 2 (-1)^{n-1} (2 n+1) P_n(2 x-1) \left(\sum _{k=n+1}^{\infty } \frac{(-1)^{k-1}}{k^2}\right)$$ can be used to compute $\sum_{n=1...
Two years ago, I discovered the following neat result:$$\tag{*}\small{ \int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x} dx = -168 \text{Li}_5\left(\frac{1}{2}\right) \zeta (3)+96 \text{Li}_4\left(\frac{1}{2}\right){}^2-\frac{19}{15} \pi ^4 \text{Li}_4\left(\frac{1}{2}\right)+\\ 12 \pi ^2 \tex...
TL;DR: I'm trying to find the first-order condition (FOC) for an optimization problem with two state variables and one control variable. I don't want the value function $V$ to appear in the FOC but can't get rid of it. Help is highly appreciated. $\def\d{\delta} \def\b{\beta} \def\g{\gamma} \def\...
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