On Gregory’s coefficients

Gregory coefficients are decreasing rational numbers 1/2, 1/12, 1/24, 19/720, ..., which are similar in their role to the Bernoulli numbers and appear in a large number of problems, especially in those related to the numerical analysis and to the number theory. They first appeared in the works of the Scottish mathematician James Gregory in 1671 and were subsequently rediscovered many times. Among the famous mathematicians who rediscovered them, we find Laplace, Mascheroni, Fontana, Bessel, Clausen, Hermite, Pearson and Fisher. Furthermore, Gregory's coefficients can rightly be considered one of the most frequently rediscovered in mathematics (the last rediscovery dates back to our century), the reason for which in literature we can find them under various names (e.g. reciprocal logarithmic numbers, Bernoulli numbers of the second kind, δ-Bernoulli numbers, Cauchy numbers, etc.). For the same reason, their authorship is often attributed to various mathematicians. In this talk, we will discuss the history of these coefficients, their discovery and subsequent rediscoveries, as well as their role in some interesting mathematical problems.