Multiple harmonic sums and Wolstenholme's theorem

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$) involves a linear combination of $n$ multiple harmonic sums, and holds $\mathrm{m}\mathrm{o}\mathrm{d}{p}^{2n+3}$. The coefficients in these congruences are integers depending on $n$ and $k$, but independent of $p$. More generally, we construct a family of congruences for $\left(\genfrac{}{}{0px}{}{kp-1}{p-1}\right)\mathrm{m}\mathrm{o}\mathrm{d}{p}^{2n+3}$, whose members contain a variable number of terms, and show that in this family there is a unique “optimized” congruence involving the fewest terms. The special case $k=2$ and $n=0$ recovers Wolstenholme’s theorem ${2p-1\choose p-1}\equiv 1\mod{p^3}$,validforallprimes$p\geq 5$\mathrm{.}Wealsocharacterizethosetriples$(n, k, p)$forwhichtheoptimizedcongruenceholdsmoduloanextrapowerof$p$:theyarepreciselythosewitheither$p$ dividing the numerator of the Bernoulli number $B_{p-2n-k}$, or $k\equiv 0,1\mathrm{m}\mathrm{o}\mathrm{d}p$.