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
||OEIS: A002206 (numerators),
OEIS: A002207 (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.
Computation and representations
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
Schröder's integral formula
or the finite summation formula
where s(n,ℓ) are the signed Stirling numbers of the first kind.
Bounds and asymptotic behavior
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.
Series with Gregory coefficients
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. More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin  and some other authors calculated
Alabdulmohsin also gives these identities
Candelperger, Coppo and Young showed that
where Hn are the harmonic numbers.
Blagouchine 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.
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
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 ψn(s) such that
and call them Bernoulli polynomials of the second kind. From the above, it is clear that Gn = ψn(0).
Carlitz generalized Jordan's polynomials ψn(s) by introducing polynomials β
Blagouchine 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
so that Gn = cn(1)/n! Numbers cn(k) are called by the author poly-Cauchy numbers. Coffey
and therefore |Gn| = Pn+1(1).
- ^ a b c d Ch. Jordan. The Calculus of Finite Differences Chelsea Publishing Company, USA, 1947.
- ^ a b L. Comtet. Advanced combinatorics (2nd Edn.) D. Reidel Publishing Company, Boston, USA, 1974.
- ^ a b H.T. Davis. The approximation of logarithmic numbers. Amer. Math. Monthly, vol. 64, no. 8, pp. 11–18, 1957.
- ^ P. C. Stamper. Table of Gregory coefficients. Math. Comp. vol. 20, p. 465, 1966.
- ^ a b D. Merlini, R. Sprugnoli, M. C. Verri. The Cauchy numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006.
- ^ a b G. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol. 14, 11.4.8, 2011
- ^ a b P.T. Young. A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers. J. Number Theory, vol. 128, pp. 2951–2962, 2008.
- ^ a b V. Kowalenko. Properties and Applications of the Reciprocal Logarithm Numbers. Acta Appl. Math., vol. 109, pp. 413–437, 2010.
- ^ a b V. Kowalenko. Generalizing the reciprocal logarithm numbers by adapting the partition method for a power series expansion. Acta Appl. Math., vol. 106, pp. 369–420, 2009.
- ^ a b c I. M. Alabdulmohsin. Summability calculus, arXiv:1209.5739, 2012.
- ^ a b c I. M. Alabdulmohsin. Summability calculus: a Comprehensive Theory of Fractional Finite Sums, Springer International Publishing, 2018.
- ^ F. Qi and X.-J. Zhang An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind. Bull. Korean Math. Soc., vol. 52, no. 3, pp. 987–98, 2015.
- ^ Weisstein, Eric W. "Logarithmic Number." From MathWorld—A Wolfram Web Resource.
- ^ a b J. C. Kluyver. Euler's constant and natural numbers. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924.
- ^ a b J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, NewYork, USA, 1950.
- ^ a b c L. Carlitz. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math., vol. 25, pp. 323–330,1961.
- ^ a b c d e f g h
Ia.V. Blagouchine. Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1. J.Math. Anal. Appl., 2015.
- ^ a b c d e 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.
- ^ a b I. V. Blagouchine, A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind, Journal of Integer Sequences, Vol. 20, No. 3 (2017), Article 17.3.8 arXiv:1612.03292
- ^ Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)
- ^ a b M.W. Coffey. Series representations for the Stieltjes constants. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014.
Ia.V. Blagouchine. A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations J. Number Theory, vol. 148, pp. 537–592 and vol. 151, pp. 276–277, 2015.
B. Candelpergher and M.-A. Coppo. A new class of identities involving Cauchy numbers, harmonic numbers and zeta values. Ramanujan J., vol. 27, pp. 305–328, 2012.
- ^ B. Candelpergher and M.-A. Coppo. A new class of identities involving Cauchy numbers, harmonic numbers and zeta values. Ramanujan J., vol. 27, pp. 305–328, 2012
- ^ OEIS: A269330
- ^ OEIS: A270857
- ^ OEIS: A270859
- ^ a b N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.
- ^ Ia.V. Blagouchine. Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only J. Number Theory, vol. 158, pp. 365–396, 2016.
- ^ M. O. Rubinstein. Identities for the Riemann zeta function Ramanujan J., vol. 27, pp. 29–42, 2012.
- ^ a b c Takao Komatsu. On poly-Cauchy numbers and polynomials, 2012.
Ia.V. Blagouchine. Three Notes on Ser's and Hasse's Representations for the Zeta-functions Integers (Electronic Journal of Combinatorial Number Theory), vol. 18A, Article #A3, pp. 1–45, 2018. arXiv:1606.02044