Beyond Wolstenholme's Theorem

Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$ or $(a+k)^m(b+k)^n(c+k)^s$ as $k$ ranges over all residues modulo $p$. We consider the sums of double or triple ratios of such terms. We showed that each of such sums is congruent to some simpler expression involving certain binomial coefficients. We also generalize these results to the sums of products or ratios of arbitrary $n$ terms: $(a_1+k)^{m_1}$, ..., $(a_n+k)^{m_n}$. We relate such summations to the sum of certain coefficients of polynomials of type $(a_1-a_n+x)^{m_1} \cdots (a_{n-1}-a_n+x)^{m_{n-1}}$.