Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,[1][2][3][4][5][6][7][8][9][10][11][12][13] are the rational numbers
that occur in the Maclaurin series expansion of the reciprocal logarithm
Gregory coefficients are alternating Gn = (−1)n−1|Gn| 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.[1][5][14][15][16][17]
More information n, ... ...
n |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
... |
OEIS sequences |
Gn |
+1/2 |
−1/12 |
+1/24 |
−19/720 |
+3/160 |
−863/60480 |
+275/24192 |
−33953/3628800 |
+8183/1036800 |
−3250433/479001600 |
+4671/788480 |
... |
OEIS: A002206 (numerators),
OEIS: A002207 (denominators) |
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The simplest way to compute Gregory coefficients is to use the recurrence formula
with G1 = 1/2.[14][18] 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 s(n,ℓ) are the signed Stirling numbers of the first kind.
and Schröder's integral formula[19][20]
The Gregory coefficients satisfy the bounds
given by Johan Steffensen.[15] These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine.[17] In particular,
Asymptotically, at large index n, these numbers behave as[2][17][19]
More accurate description of Gn at large n may be found in works of Van Veen,[18] Davis,[3] Coffey,[21] Nemes[6] and Blagouchine.[17]
Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include
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.[17][22] More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko,[8] Alabdulmohsin [10][11] and some other authors calculated
Alabdulmohsin[10][11] also gives these identities with
Candelperger, Coppo[23][24] and Young[7] showed that
where Hn are the harmonic numbers.
Blagouchine[17][25][26][27] provides the following identities
where li(z) 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.[1][17][18][28][29]
Various generalizations are possible for the Gregory coefficients. Many of them may be obtained by modifying the parent generating equation. For example, Van Veen[18] consider
and hence
Equivalent generalizations were later proposed by Kowalenko[9] and Rubinstein.[30] In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers
see,[18][28] so that
Jordan[1][16][31] defines polynomials ψn(s) such that
and call them Bernoulli polynomials of the second kind. From the above, it is clear that Gn = ψn(0).
Carlitz[16] generalized Jordan's polynomials ψn(s) by introducing polynomials β
and therefore
Blagouchine[17][32] introduced numbers Gn(k) such that
obtained their generating function and studied their asymptotics at large n. Clearly, Gn = Gn(1). These numbers are strictly alternating Gn(k) = (-1)n-1|Gn(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[31]
so that Gn = cn(1)/n! Numbers cn(k) are called by the author poly-Cauchy numbers.[31] Coffey[21]
defines polynomials
and therefore |Gn| = Pn+1(1).
Ch. Jordan. The Calculus of Finite Differences Chelsea Publishing Company, USA, 1947.
L. Comtet. Advanced combinatorics (2nd Edn.) D. Reidel Publishing Company, Boston, USA, 1974.
P. C. Stamper. Table of Gregory coefficients. Math. Comp. vol. 20, p. 465, 1966.
G. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol. 14, 11.4.8, 2011
J. C. Kluyver. Euler's constant and natural numbers. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924.
J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, New York, USA, 1950.
L. Carlitz. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math., vol. 25, pp. 323–330,1961.
S.C. Van Veen. Asymptotic expansion of the generalized Bernoulli numbers Bn(n − 1) for large values of n (n integer). Indag. Math. (Proc.), vol. 13, pp. 335–341, 1951.
Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)
N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.