Gregory coefficients , also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,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.D. Merlini, R. Sprugnoli, M. C. Verri. The Cauchy numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006.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.V. Kowalenko. Properties and Applications of the Reciprocal Logarithm Numbers. Acta Appl. Math., vol. 109, pp. 413–437, 2010.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.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. 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.J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, New York, USA, 1950.

Numerical values

OEIS sequences (numerators), (denominators)

Computation and representations The simplest way to compute Gregory coefficients is to use the recurrence formula

with .J. C. Kluyver. Euler's constant and natural numbers. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924. 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 formulaI. 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:1612.03292Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)

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.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 −1. J.Math. Anal. Appl., 2015. In particular,

Asymptotically, at large index , these numbers behave as

More accurate description of at large may be found in works of Van Veen,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. Davis,H.T. Davis. The approximation of logarithmic numbers. Amer. Math. Monthly, vol. 64, no. 8, pp. 11–18, 1957. Coffey,M.W. Coffey. Series representations for the Stieltjes constants. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014. NemesG. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol. 14, 11.4.8, 2011 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 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, CoppoB. 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 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.N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.Ia.V. Blagouchine. Expansions of generalized Euler's constants into the series of polynomials in −2 and into the formal enveloping series with rational coefficients only. J. Number Theory, vol. 158, pp. 365–396, 2016.

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 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 . CarlitzL. Carlitz. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math., vol. 25, pp. 323–330,1961. generalized Jordan's polynomials by introducing polynomials

and therefore

BlagouchineIa.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 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 .

See also Stirling polynomials Bernoulli polynomials of the second kind

References