Inversions Relating Stirling, Tanh, Lah Numbers and an Application to Mathematical Statistics
نویسنده
چکیده
Abstract. 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.
منابع مشابه
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The recurrence is significant since if we set ai = 0, bi = i for all i we get the Stirling numbers of the second kind, and if we set ai = bi = i for all i we get the Lah numbers. The Lah numbers count the partitions of an n-set into blocks of size k in which the numbers in each block are ordered. The proof in [1] involves generating functions and is rather long. Here we give a short induction p...
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