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Let X(μ)={Xt(μ);t≥0}$X^{(\mu )} = \{ X_t^{(\mu )} ;t \ge 0\} $, μ> 0, be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space ℍnhaving the Laplace–Beltrami operator with drift as its generator. We prove the reflection principle for X(μ)which enables us to study the process X(μ)killed when exiting the hyperbolic half-space, that is the set D= {x∈ ℍn: x1> 0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of Dfor the process X(μ). Finally, we derive formula for the λ-Poisson kernel of the set D.

In the original article there were capitalization errors in two of the references. The corrected references follow:

We derive new congruences bounding r-Bell and derangement polynomials, which generalize the existing ones, while the presented approach is significantly simpler and, at the same time, more informative. Namely, we provide precise identities that imply the congruences and explain somehow their nature.