Gregory coefficients
Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers
(denominators)
that occur in the Maclaurin series expansion of the reciprocal logarithm
Gregory coefficients are alternating Gn = (−1)n−1|Gn| and decreasing in absolute value. These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many famous mathematicians and often appear in works of modern authors who do not recognize them.
The simplest way to compute Gregory coefficients is to use the recurrence formula
with G1 = 1/2. Gregory coefficients may be also computed explicitly via the following differential
the integral
Schröder's integral formula
or the finite summation formula
where s(n,ℓ) are the signed Stirling numbers of the first kind.
The Gregory coefficients satisfy the bounds
given by Johan Steffensen. These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine. In particular,
Asymptotically, at large index n, these numbers behave as
More accurate description of Gn at large n may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine.
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