The universal Kummer congruences

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number $\frac{{\widehat{B}}_n}{n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers $\frac{{\widehat{B}}_n}{n}$ when $n$ is divisible by $p-1$.