An alternating Euler Sum

In this post, we will evaluate the famous Euler sum: $\sum_{n=1}^\infty (-1)^{n+1}\frac{H_n}{n^3}\quad \color{blue}{\cdots (*)}$ where $H_n = \sum_{k=1}^n \frac{1}{k}=\int_0^1 \frac{1-t^n}{1-t}dt$ is the $n$-th harmonic number. This series resists contour integration techniques which makes it’s computation quite challenging. We will start with the following well known generating function identity: $\sum_{n=1}^\infty H_n t^n = -\frac{\log(1-t)}{1-t} ,\quad -1 \leq t < 1$ Dividing the above equation by $t$ and integrating both sides, gives us: $\begin{aligned} \sum_{n=1}^\infty H_n \frac{x^n}{n} &= -\int_0^x\frac{\log(1-t)}{t(1-t)}dt \\ &= -\int_0^x \left(\frac{1}{t} + \frac{1}{1-t} \right)\log(1-t) \; dt \\ &= \text{Li}_2(x) + \frac{1}{2}\log^2(1-x) \end{aligned}$ where $-1\leq x < 1$.

1. Computation of $\sum_{n=1}^\infty \frac{H_n}{n^2}x^n$

Now, we will use the same process to evaluate $\sum_{n=1}^\infty \frac{H_n}{n^2}x^n$ but we’ll have to divide the evaluation into two parts.

(I) First consider the case when $0 \leq x < 1$. Actually, we don’t need this case for evaluating (*) but I’ll do it anyway for completeness. We have: $\begin{aligned} \sum_{n=1}^\infty \frac{H_n}{n^2} x^n &= \int_0^x\left(\frac{\text{Li}_2(t)}{t} + \frac{1}{2}\frac{\log^2(1-t)}{t}\right)dt \\ &= \text{Li}_3(x) + \frac{1}{2}\int_0^x \frac{\log^2(1-t)}{t} dt \\ &= \text{Li}_3(x) + \frac{1}{2}\int_{\log(1-x)}^0 \frac{t^2 e^t}{1-e^t}dt \quad (t\mapsto 1-e^t) \\ &= \text{Li}_3(x) + \frac{1}{2}\sum_{n=1}^\infty \int_{\log(1-x)}^0 t^2 e^{nt} \; dt \\ &= \text{Li}_3(x) + \frac{1}{2}\sum_{n=1}^\infty \left(\frac{2}{n^3} -\frac{\log^2(1-x)}{n}(1-x)^n+\frac{2\log(1-x)}{n^2}(1-x)^n-2\frac{(1-x)^n}{n^3}\right) \\ &= \text{Li}_3(x) + \frac{\log^2(1-x)\log(x)}{2} + \log(1-x)\text{Li}_2(1-x) - \text{Li}_3(1-x) + \zeta(3) \quad\color{blue}{\cdots (1)} \end{aligned}$ (II) Similarly, for the case $-1\leq x < 0$, we obtain:

$\begin{aligned} \sum_{n=1}^\infty \frac{H_n}{n^2}x^n &= \text{Li}_3(x) -\frac{1}{2}\int_x^0 \frac{\log^2(1-t)}{t}dt \\ &= \text{Li}_3(x) +\frac{1}{2}\int_{0}^{-x} \frac{\log^2(1+t)}{t}dt \quad (t\mapsto -t) \\ &= \text{Li}_3(x) +\frac{1}{2}\int_{0}^{\log(1-x)} \frac{t^2 e^t}{e^t-1}dt \quad (t\mapsto e^t-1) \\ &= \text{Li}_3(x) +\frac{1}{2} \sum_{n=0}^\infty \int_0^{\log(1-x)} t^2 e^{-nt} dt \\ &= \text{Li}_3(x) +\frac{\log^3(1-x)}{6} + \frac{1}{2}\sum_{n=1}^\infty \int_0^{\log(1-x)} t^2 e^{-nt} dt \\ &= \text{Li}_3(x) +\frac{\log^3(1-x)}{6} + \frac{1}{2}\sum_{n=1}^\infty \left(\frac{2}{n^3} -\log^2(1-x)\frac{(1-x)^{-n}}{n}-2\log(1-x)\frac{(1-x)^{-n}}{n^2}-2\frac{(1-x)^{-n}}{n^3}\right) \\ &= \text{Li}_3(x) -\frac{\log^3(1-x)}{3} +\frac{\log^2(1-x)\log(-x)}{2} -\log(1-x)\text{Li}_2\left( \frac{1}{1-x}\right)-\text{Li}_3\left(\frac{1}{1-x}\right)+\zeta(3) \\ &\quad\color{blue}{\cdots (2)} \end{aligned}$

2. Evaluation of $\sum_{n=1}^\infty \frac{H_n}{n^3}x^n$

This is the hardest step in the evaluation. Once again, we will divide the evaluation into two parts.

(I) Like section 1, we’ll first consider the case when $0\leq x < 1$.

Using integration by parts, we obtain: $\begin{aligned} \int_0^x \frac{\zeta(3)-\text{Li}_3(1-t)}{t}dt &= \log(t)\left(\zeta(3)-\text{Li}_3(1-t) \right) \Big|_{0}^x - \int_0^x \frac{\log(t)\text{Li}_2(1-t)}{1-t}dt \\ &= \log(x)\left(\zeta(3)-\text{Li}_3(1-x) \right) - \frac{1}{2}\text{Li}_2^2 (1-x) + \frac{\zeta^2 (2)}{2} \quad \color{blue}{\cdots (3)} \end{aligned}$

$\begin{aligned} \int_0^x \frac{\log(1-t)\text{Li}_2(1-t)}{t}dt &= \log(t)\log(1-t)\text{Li}_2(1-t) \Big|_0^x - \int_0^x \log(t) \left(\frac{-\text{Li}_2(1-t)}{1-t} + \frac{\log(1-t)\log(t)}{1-t}\right)dt \\ &= \log(x)\log(1-x)\text{Li}_2(1-x) + \frac{1}{2}\text{Li}_2^2(1-x)-\frac{\zeta^2(2)}{2} - \int_0^x \frac{\log^2(t) \log(1-t)}{1-t}dt \\ &\quad \color{blue}{\cdots (4)} \end{aligned}$

$\begin{aligned} \int_0^x \frac{\log(t)\log^2(1-t)}{2t}dt &= \frac{1}{4}\log^2(t)\log^2(1-t)\Big|_0^x +\frac{1}{2}\int_0^x \frac{\log^2(t)\log(1-t)}{1-t}dt \\ &= \frac{1}{4}\log^2(x)\log^2(1-x) +\frac{1}{2}\int_0^x \frac{\log^2(t)\log(1-t)}{1-t}dt \quad \color{blue}{\cdots (5)} \end{aligned}$

Putting together the results of equations (3), (4) and (5) gives us: $\begin{aligned} \sum_{n=1}^\infty \frac{H_n}{n^3}x^n &= \text{Li}_4(x) +\frac{1}{4}\log^2(x)\log^2(1-x) + \log(x)\log(1-x)\text{Li}_2(1-x)\\ &\quad + \log(x)\left(\zeta(3)-\text{Li}_3(1-x) \right) -\frac{1}{2}\int_0^x \frac{\log^2(t)\log(1-t)}{1-t}dt \quad \color{blue}{\cdots (6)} \end{aligned}$

(II) Now, let $-1\leq x < 0$. We have: $\begin{aligned} -\int_x^0 \frac{\zeta(3)-\text{Li}_3\left(\frac{1}{1-t}\right)}{t}dt &= \int_0^{\frac{-x}{1-x}}\frac{\zeta(3)-\text{Li}_3(1-t)}{t(1-t)}dt \quad \left(t\mapsto\frac{-t}{1-t} \right)\\ &= \int_0^{\frac{-x}{1-x}}\frac{\zeta(3)-\text{Li}_3(1-t)}{t}dt + \int_0^{\frac{-x}{1-x}}\frac{\zeta(3)-\text{Li}_3(1-t)}{1-t}dt \\ &= \log\left(\frac{-x}{1-x}\right)\left(\zeta(3) - \text{Li}_3\left(\frac{1}{1-x}\right) \right) - \frac{1}{2}\text{Li}_2^2\left(\frac{1}{1-x}\right) + \frac{\zeta^2(2)}{2} \\ &\quad + \zeta(3)\log(1-x) + \text{Li}_4\left(\frac{1}{1-x} \right) - \zeta(4) \quad \color{blue}{\cdots (7)} \end{aligned}$ Here, we used equation (3) to simplify the integral after the transformation. Similarly, we obtain: $\begin{aligned} \int_x^0 \frac{\log(1-t)\text{Li}_2\left(\frac{1}{1-t}\right)}{t}dt &= \int_0^{\frac{-x}{1-x}}\frac{\log(1-t)\text{Li}_2(1-t)}{t(1-t)}dt \\ &= \int_0^{\frac{-x}{1-x}}\frac{\log(1-t)\text{Li}_2(1-t)}{t}dt + \int_0^{\frac{-x}{1-x}}\frac{\log(1-t)\text{Li}_2(1-t)}{1-t}dt \\ &= -\log\left(\frac{-x}{1-x} \right)\log(1-x)\text{Li}_2\left(\frac{1}{1-x}\right)+\frac{1}{2}\text{Li}_2^2\left(\frac{1}{1-x}\right)-\frac{\zeta^2(2)}{2} \\ &\quad +\text{Li}_4\left(\frac{1}{1-x}\right)+\log(1-x)\text{Li}_3\left(\frac{1}{1-x}\right) -\zeta(4) \\ &\quad -\int_0^{\frac{-x}{1-x}}\frac{\log^2(t)\log(1-t)}{1-t}dt \quad \color{blue}{\cdots (8)} \end{aligned}$

$\begin{aligned} \int_x^0 \left( \frac{\log^3(1-t)}{3t} - \frac{\log^2(1-t)\log(-t)}{2t}\right) dt &= \frac{1}{6}\log^3(1-x)\log\left(\frac{-x}{1-x}\right) + \frac{1}{24}\log^4(1-x) \\ &\quad +\frac{1}{4}\log^2\left(\frac{-x}{1-x}\right)\log^2(1-x) + \frac{1}{2}\int_0^{\frac{-x}{1-x}}\frac{\log^2(t)\log(1-t)}{1-t}dt \\ &\quad \color{blue}{\cdots (9)} \end{aligned}$

Finally, putting these results together gives:

$\begin{aligned} \sum_{n=1}^\infty \frac{H_n}{n^3}x^n &= \text{Li}_4(x) + 2\text{Li}_4\left( \frac{1}{1-x}\right) + \log\left(\frac{-x}{1-x} \right)\left(-\text{Li}_3\left(\frac{1}{1-x}\right) +\frac{\log^3(1-x)}{6}-\log(1-x)\text{Li}_2\left(\frac{1}{1-x} \right)\right) \\ &\quad + \log(1-x)\text{Li}_3\left(\frac{1}{1-x}\right) + \frac{\log^4(1-x)}{24} +\frac{1}{4}\log^2 \left(\frac{-x}{1-x}\right)\log^2(1-x) +\zeta(3) \log(-x)- 2\zeta(4) \\ &\quad -\frac{1}{2}\int_0^{\frac{-x}{1-x}}\frac{\log^2(t)\log(1-t)}{1-t}dt \quad \color{blue}{\cdots (10)} \end{aligned}$

The integral $\int \frac{\log^2(t)\log(1-t)}{1-t}dt$ can be evaluated in terms of Polylogarithms (refer section 7.6 of “Polylogarithms and Associated Functions” by Leonard Lewin). But it turns out that for evaluating (*) such a calculation is not needed.

3. Evaluation of $\int_0^{\frac{1}{2}}\frac{\log^2(t)\log(1-t)}{1-t}dt$

Let $I=\int_0^{1}\frac{\log^2(t)\log(1-t)}{1-t}dt$ and $J=\int_0^{\frac{1}{2}}\frac{\log^2(t)\log(1-t)}{1-t}dt$. The integral, $I$, can be evaluated by the differentiation under the integral technique. $\begin{aligned} I &= \lim_{y\to 0^+}\lim_{x\to 1}\frac{\partial^2 }{\partial x^2} \frac{\partial }{\partial y} \int_0^1 t^{x-1} (1-t)^{y-1} dt \\ &= \lim_{y\to 0^+}\lim_{x\to 1}\frac{\partial^2 }{\partial x^2} \frac{\partial }{\partial y} \left\{\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \right\} \end{aligned}$ It takes a bit of effort to compute the derivatives in terms of Polygamma functions so I won’t write the details here. One can verify that the end result is: $I = -\frac{\pi^4}{180}$ Now, apply integration by parts to the integral $J$: $\begin{aligned} J &= \int_0^{\frac{1}{2}}\frac{\log^2(t)\log(1-t)}{1-t}dt \\ &= -\frac{\log^2(t)\log^2(1-t)}{2}\Big|_0^{\frac{1}{2}} + \int_0^{\frac{1}{2}} \frac{\log(t)\log^2(1-t)}{t}dt \\ &= -\frac{\log^4(2)}{2} + \int_{\frac{1}{2}}^1 \frac{\log^2(t)\log(1-t)}{1-t}dt \quad (t\mapsto 1-t)\\ &= -\frac{\log^4(4)}{2} + I-J \end{aligned}$ Now, one can solve the above equation for $J$ to get: $\begin{aligned} J &= -\frac{\log^4(2)}{4}+\frac{I}{2} \\ &= -\frac{\log^4(2)}{4}-\frac{\pi^4}{360} \end{aligned}$

Finally, we have every thing needed to evaluate (*). Plugging in $x=-1$ in equation (10) and performing some simplifications gives us the beautiful result: $\boxed{\sum_{n=1}^\infty \frac{H_n}{n^3}(-1)^{n+1} = \frac{11\pi^4}{360}+\frac{\pi^2}{12}\log^2(2)-\frac{\log^4(2)}{12}-\frac{7}{4}\log(2)\zeta(3) -2\text{Li}_4\left(\frac{1}{2}\right)}$ A similar result is obtained by plugging $x=\frac{1}{2}$ into equation (6): $\boxed{\sum_{n=1}^\infty \frac{H_n}{n^3 2^n}= \frac{\pi^4}{720}+\frac{\log^4(2)}{24}-\frac{1}{8}\log(2)\zeta(3) +\text{Li}_4\left(\frac{1}{2}\right)}$

4. Further Results

Let $H_n^{(2)} = \zeta(2) - \psi_1(n+1) = \sum_{k=1}^n \frac{1}{k^2}$. Then, it is easy to verify that: $\sum_{n=1}^\infty t^n H_n^{(2)} = \frac{\text{Li}_2(t)}{1-t}, \quad -1\leq t < 1$ Now, dividing the above equation by $t$ and integrating both sides, gives us: $\begin{aligned} \sum_{n=1}^\infty \frac{H_n^{(2)}}{n}x^n &= \int_0^x \text{Li}_2(t)\left( \frac{1}{t}+\frac{1}{1-t}\right) \\ &= \text{Li}_3(x) + \int_0^x \frac{\text{Li}_2(t)}{1-t} dt \\ &= \text{Li}_3(x) -\log(1-x)\text{Li}_2(x) - \int_0^x \frac{\log^2(1-t)}{t}dt \end{aligned}$ From our previous calculation, we know that $\int_0^x \frac{\log^2(1-t)}{t}dt = 2\sum_{n=1}^\infty \frac{H_n}{n^2}x^n -2\text{Li}_3(x)$ Hence, we obtain the following relation: $\sum_{n=1}^\infty \frac{H_n^{(2)}}{n}x^n =3\text{Li}_3(x)-\log(1-x)\text{Li}_2(x) - 2\sum_{n=1}^\infty \frac{H_n}{n^2}x^n \quad \color{blue}{\cdots (11)}$ Once again, divide the above equation by $x$ and integrate both sides to get: $\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^2}x^n = 3\text{Li}_4(x) + \frac{1}{2}\text{Li}_2^2(x) - 2\sum_{n=1}^\infty \frac{H_n}{n^3}x^n \quad \color{blue}{\cdots (12)}$ Equations (11) and (12) are valid for $-1\leq x < 1$. Plugging in $x=\frac{1}{2}$ in equation (12) and using the known closed form for $\sum_{n=1}^\infty \frac{H_n}{n^3 2^n}$ gives us: $\boxed{ \sum_{n=1}^\infty \frac{H_n^{(2)}}{n^2 2^n} = \frac{\pi^4}{1440}-\frac{\pi^2}{24}\log^2(2) + \frac{\log^4(2)}{24}+ \frac{1}{4}\log(2)\zeta(3) + \text{Li}_4\left(\frac{1}{2}\right) }$