## $$p$$-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.)

### Keywords:

special numbers; $$p$$-adic analysis; congruences
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### References:

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