  Inversion formulas have been found, converting between Stirling, tanh and Lah
numbers. Tanh and Lah polynomials, analogous to the Stirling polynomials, have
been defined and their basic properties established. New identities for
Stirling and tangent numbers and polynomials have been derived from the general
inverse relations. In the second part of the paper, it has been shown that if
shifted-gamma probability densities and negative binomial distributions are
matched by equating their first three semi-invariants (cumulants), then the
cumulants of the two distributions are related by a pair of reciprocal linear
combinations equivalent to the inversion formulas established in the first
part.
