On some connections between the Gompertz function and special numbers
Grzegorz Rz\k{a}dkowski, Wojciech Rz\k{a}dkowski, Pawe{\l} W\'ojcicki
TL;DR
This paper explores mathematical connections between the Gompertz function, certain probability distributions, and special numbers like Stirling and Bernoulli numbers, revealing new formulas and relationships.
Contribution
It establishes novel links between the Gompertz function, Fisher-Tippett, Gumbel distributions, and special numbers, including an analog of the Grosset-Veselov formula.
Findings
Connected Gumbel distribution to Bernoulli numbers.
Derived an analog of the Grosset-Veselov formula.
Linked special functions with combinatorial numbers.
Abstract
In the present paper we show that the Gompertz function, the Fisher-Tippett and the Gumbel probability distributions are related to both Stirling numbers of the second kind and Bernoulli numbers. Especially we prove for the Gumbel probability density function an analog of the Grosset-Veselov formula which connects 1-soliton solution of the KdV equation with Bernoulli numbers.
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