Gregory coefficients "G""n", 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
formula_1
Gregory coefficients are alternating "G""n" 
 (−1)"n"−1|"G""n"| for "n" ; 0 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.
Computation and representations.
The simplest way to compute Gregory coefficients is to use the recurrence formula
 formula_2
with "G"1 
 . Gregory coefficients may be also computed explicitly via the following differential
 formula_3
or the integral
 formula_4
which can be proved by integrating formula_5 between 0 and 1 with respect to formula_6, once directly and the second time using the binomial series expansion first.
It implies the finite summation formula
 formula_7
where "s"("n","ℓ") are the signed Stirling numbers of the first kind.
and Schröder's integral formula
 formula_8
Bounds and asymptotic behavior.
The Gregory coefficients satisfy the bounds
formula_9
given by Johan Steffensen. These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine. In particular,
formula_10
Asymptotically, at large index "n", these numbers behave as
formula_11
More accurate description of "G""n" at large "n" may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine.
Series with Gregory coefficients.
Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include
formula_12
where "γ" 
 0.5772156649... 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
formula_13
Alabdulmohsin also gives these identities with
formula_14
Candelperger, Coppo and Young showed that
formula_15
where "H""n" are the harmonic numbers.
Blagouchine provides the following identities
formula_16
where li("z") is the integral logarithm and formula_17 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.
Generalizations.
Various generalizations are possible for the Gregory coefficients. Many of them may be obtained by modifying the parent generating equation. For example, Van Veen consider
formula_18
and hence
 formula_19
Equivalent generalizations were later proposed by Kowalenko and Rubinstein. In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers
 formula_20
see, so that
 formula_21
Jordan defines polynomials "ψn"("s") such that
 formula_22
and call them "Bernoulli polynomials of the second kind". From the above, it is clear that "G""n" 
 "ψn"(0).
Carlitz generalized Jordan's polynomials "ψ""n"("s") by introducing polynomials "β"
formula_23
and therefore
 formula_24
Blagouchine introduced numbers "G""n"("k") such that
 formula_25
obtained their generating function and studied their asymptotics at large "n". Clearly, "G""n" 
 "G""n"(1). These numbers are strictly alternating "G""n"("k") 
 (-1)"n"-1|"G""n"("k")| 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 
 formula_26
so that "G""n" 
 "c""n"(1)/"n"! Numbers "c""n"("k") are called by the author "poly-Cauchy numbers". Coffey
defines polynomials
 formula_27
and therefore |"G""n"| 
 "P""n"+1(1).
References.
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