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\(p\)-adic analysis and congruences of the coefficients of the series \(e^{xe^ x}\). (Analyse \(p\)-adique et congruences des coefficients de la série \(e^{xe^ x}\).) (French) Zbl 0871.11014

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 \(h\geq 1\), \(n\geq 2^{2h}(h+2^{-1})\), \[ B_{n+2^{3h+1}}\equiv B_n\bmod(2^h). \]

MSC:

11B83 Special sequences and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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References:

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