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\(p\)-adic analysis and Bell numbers of two variables. (Analyse \(p\)-adique et nombres de Bell à deux variables.) (French) Zbl 0866.11014

The Bernoulli numbers of two arguments, of order \(n\), are defined by \[ B(n,a,b)= \sum^n_{k=0}R(n,k,a)\cdot b^k, \qquad R(n,k,a)={1\over k!}\sum^k_{j=0} (-1)^{k-j}\cdot{k\choose j}(a+j)^n. \] These numbers contain, as a special case, generalized Bell numbers considered by Carlitz and Chinthayama and Gandhi.
The author shows that the generating function of the Bell numbers of two variables is a \(p\)-adic analytical element on a quasi-connected domain of \(\mathbb{C}_p\). By the classical theorems of \(p\)-adic analysis, the above result is equivalent to certain congruences for the Bell numbers. Certain theorems by Radoux, Carlitz, Layman, Layman and Prather are also generalized. The explicit statements of these results are too complicated to be stated here.

MSC:

11B73 Bell and Stirling numbers
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
05A18 Partitions of sets
30G06 Non-Archimedean function theory
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