New harmonic number identities with applications

We determine the explicit formulas for the sum of products of homogeneous multiple harmonic sums $\sum_{k=1}^n \prod_{j=1}^r H_k(\{1\}^{\lambda_j})$ when $\sum_{j=1}^r \lambda_j\leq 5$. We apply these identities to the study of two congruences modulo a power of a prime

The dinner table problem: the rectangular case

$n$ people are seated randomly at a rectangular table with $\lfloor n/2\rfloor$ and $\lceil n/2\rceil$ seats along the two opposite sides for two dinners. What's the probability that neighbors at the first dinner are no more neighbors at the second one? We give an explicit formula and we show that its asymptotic behavior as $n$ goes to infinity is $e^{-2}(1+4/n)$ (it is known that it is $e^{-2}(1-4/n)$ for a round table). A more general permutation problem is also considered.Comment: 10 page

Supercongruences for a truncated hypergeometric series

The purpose of this note is to obtain some congruences modulo a power of a prime $p$ involving the truncated hypergeometric series $$\sum_{k=1}^{p-1} {(x)_k(1-x)_k\over (1)_k^2}\cdot{1\over k^a}$$ for $a=1$ and $a=2$. In the last section, the special case $x=1/2$ is considered.Comment: Revised versio

Supercongruences for the Almkvist-Zudilin numbers

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$; while the latter (essentially) focuses on the maximal powers $r$ and $t$ such that $c(p^rn)$ is congruent to $c(p^{r-1}n)$ modulo $p^t$. This is called supercongruence. In this paper, we prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.Comment: 15 page

Congruences for central binomial sums and finite polylogarithms

We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$