Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm
Gregory coefficients are alternating for 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 mathematicians and often appear in works of modern authors, who do not always recognize them.
The simplest way to compute Gregory coefficients is to use the recurrence formula
with . Gregory coefficients may be also computed explicitly via the following differential
or the integral
which can be proved by integrating between 0 and 1 with respect to , once directly and the second time using the binomial series expansion first.
It implies the finite summation formula
where are the signed Stirling numbers of the first kind.
and Schröder's integral formula
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 , these numbers behave as
More accurate description of at large may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine.
Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include
where is Euler's constant. These results are very old, and their history may be traced back to the works of Gregorio Fontana and Lorenzo Mascheroni. More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin and some other authors calculated
Alabdulmohsin also gives these identities with
Candelperger, Coppo and Young showed that
where are the harmonic numbers. Blagouchine provides the following identities
where is the integral logarithm and is the binomial coefficient. It is also known that the zeta function, the gamma function, the polygamma functions, the Stieltjes constants and many other special functions and constants may be expressed in terms of infinite series containing these numbers.
Various generalizations are possible for the Gregory coefficients. Many of them may be obtained by modifying the parent generating equation. For example, Van Veen considered
and hence
Equivalent generalizations were later proposed by Kowalenko and Rubinstein. In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers
see, so that
Jordan defines polynomials such that
and call them Bernoulli polynomials of the second kind. From the above, it is clear that . Carlitz generalized Jordan's polynomials by introducing polynomials
and therefore
Blagouchine introduced numbers such that
obtained their generating function and studied their asymptotics at large . Clearly, . These numbers are strictly alternating and involved in various expansions for the zeta-functions, Euler's constant and polygamma functions. A different generalization of the same kind was also proposed by Komatsu
so that Numbers are called by the author poly-Cauchy numbers. Coffey defines polynomials
and therefore .