Logarithmic integrals with applications to BBP and Euler-type sums

For real numbers $p,q>1$ we consider the following family of integrals: \begin{equation*} \int_{0}^{1}\frac{(x^{q-2}+1)\log\left(x^{mq}+1\right)}{x^q+1}{\rm d}x \quad \mbox{and}\quad \int_{0}^{1}\frac{(x^{pt-2}+1)\log\left(x^t+1\right)}{x^{pt}+1}{\rm d}x. \end{equation*} We evaluate these integrals for all $m\in\mathbb{N}$, $q=2,3,4$ and $p=2,3$ explicitly. They recover some previously known integrals. We also compute many integrals over the infinite interval $[0,\infty)$. Applying these results we offer many new Euler- BBP- type sums.Comment: Accepted for publication in the Bulletin of Malaysian Mathematical Sciences Societ

On some Binomial Coefficient Identities with Applications

We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k &=\sum_{k=0}^{n}(-1)^{n+k}\binom{\beta-\alpha+n}{n-k}\binom{\beta+k}{k}(x+1)^k, \end{align*} where $n$ is a non-negative integer and $\alpha$ and $\beta$ are complex numbers, which are not negative integers. Our approach is based on a particularly interesting combination of the Taylor theorem and the Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to Alzer and Kouba, and offer a new binomial sum identity. Furthermore, as applications, we give many harmonic number sum identities. As examples, we prove that \begin{equation*} H_n=\frac{1}{2}\sum_{k=1}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{k}H_k \end{equation*} and \begin{align*} \sum_{k=0}^{n}\binom{n}{k}^2H_kH_{n-k}=\binom{2n}{n} \left((H_{2n}-2H_n)^2+H_{n}^{(2)}-H_{2n}^{(2)}\right). \end{align*}Comment: Submitte