\(p\)-adic analysis and congruences of the coefficients of the series \(e^{xe^ x}\) (Q677874)

The coefficients \((B_n)\) are defined by \(\sum^\infty_{n=0}B_n x^n/n!=e^{xe^x}\). \(B_0=1\) and \(B_n=\sum^n_{j=1}j^{(n-j)}\cdot{n\choose j}\) for \(n\geq 1\). By using methods of \(p\)-adic analysis (e.g. the \(p\)-adic Mittag-Leffler theorem or Cauchy's inequalities for the \(p\)-adic analytical elements on a quasi-connected domain of \(\mathbb{C}_p\)), the author proves the following congruences: If \(p\) is an odd prime, then for \(h\geq1\), \(n\geq p^{2h}(1+h+p^{-1})\), \[ B_{n+(p-1)p^{3h}}\equiv B_n\bmod (p^h); \] for \(h\geq 1\), \(n\geq 2^{2h}(h+2^{-1})\), \[ B_{n+2^{3h+1}}\equiv B_n\bmod(2^h). \]