Gregory coefficients

Gregory coefficients G_n, 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

${\displaystyle {\begin{aligned}{\frac {z}{\ln(1+z)}}&=1+{\frac {1}{2}}z-{\frac {1}{12}}z^{2}+{\frac {1}{24}}z^{3}-{\frac {19}{720}}z^{4}+{\frac {3}{160}}z^{5}-{\frac {863}{60480}}z^{6}+\cdots \\&=1+\sum _{n=1}^{\infty }G_{n}z^{n}\,,\qquad |z|<1\,.\end{aligned}}}$

Gregory coefficients are alternating G_n = (−1)ⁿ⁻¹|G_n| 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]

Numerical values

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)

Computation and representations

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

${\displaystyle {\frac {G_{1}}{n}}-{\frac {G_{2}}{n-1}}+{\frac {G_{3}}{n-2}}-\cdots +(-1)^{n-1}{\frac {G_{n}}{1}}\,=\,{\frac {1}{n+1}}\,,\qquad n=2,3,4,\ldots }$

with G₁ = 1/2.[14][18] Gregory coefficients may be also computed explicitly via the following differential

${\displaystyle G_{n}={\frac {1}{n!}}\left[{\frac {d^{n}}{dz^{n}}}{\frac {z}{\ln(1+z)}}\right]_{z=0},}$

the integral

${\displaystyle G_{n}={\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx,}$

Schröder's integral formula[19][20]

${\displaystyle G_{n}=(-1)^{n-1}\int _{0}^{\infty }{\frac {dx}{(1+x)^{n}(\ln ^{2}x+\pi ^{2})}},}$

or the finite summation formula

${\displaystyle G_{n}={\frac {1}{n!}}\sum _{\ell =1}^{n}{\frac {s(n,\ell )}{\ell +1}},}$

where s(n,ℓ) are the signed Stirling numbers of the first kind.

Bounds and asymptotic behavior

The Gregory coefficients satisfy the bounds

${\displaystyle {\frac {1}{6n(n-1)}}<{\big |}G_{n}{\big |}<{\frac {1}{6n}},\qquad n>2,}$

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,

${\displaystyle {\frac {\,1\,}{\,n\ln ^{2}\!n\,}}\,-\,{\frac {\,2\,}{\,n\ln ^{3}\!n\,}}\leqslant \,{\big |}G_{n}{\big |}\,\leqslant \,{\frac {\,1\,}{\,n\ln ^{2}\!n\,}}-{\frac {\,2\gamma \,}{\,n\ln ^{3}\!n\,}}\,,\qquad \quad n\geqslant 5\,.}$

Asymptotically, at large index n, these numbers behave as[2][17][19]

${\displaystyle {\big |}G_{n}{\big |}\sim {\frac {1}{n\ln ^{2}n}},\qquad n\to \infty .}$

More accurate description of G_n at large n may be found in works of Van Veen,[18] Davis,[3] Coffey,[21] Nemes[6] and Blagouchine.[17]

Series with Gregory coefficients

Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include

${\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\big |}G_{n}{\big |}=1\\[2mm]\sum _{n=1}^{\infty }G_{n}={\frac {1}{\ln 2}}-1\\[2mm]\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n}}=\gamma ,\end{aligned}}}$

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

${\displaystyle {\begin{array}{l}\displaystyle \sum _{n=2}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n-1}}=-{\frac {1}{2}}+{\frac {\ln 2\pi }{2}}-{\frac {\gamma }{2}}\\[6mm]\displaystyle \displaystyle \sum _{n=1}^{\infty }\!{\frac {{\big |}G_{n}{\big |}}{n+1}}=1-\ln 2.\end{array}}}$

Alabdulmohsin[10][11] also gives these identities

${\displaystyle {\begin{aligned}&{\big |}G_{1}{\big |}+{\big |}G_{2}{\big |}-{\big |}G_{4}{\big |}-{\big |}G_{5}{\big |}+{\big |}G_{7}{\big |}+{\big |}G_{8}{\big |}-{\big |}G_{10}{\big |}-{\big |}G_{11}{\big |}+\cdots ={\frac {\sqrt {3}}{\pi }}\\[2mm]&{\big |}G_{2}{\big |}+{\big |}G_{3}{\big |}-{\big |}G_{5}{\big |}-{\big |}G_{6}{\big |}+{\big |}G_{8}{\big |}+{\big |}G_{9}{\big |}-{\big |}G_{11}{\big |}-{\big |}G_{12}{\big |}+\cdots ={\frac {2{\sqrt {3}}}{\pi }}-1\\[2mm]&{\big |}G_{1}{\big |}-{\big |}G_{3}{\big |}-{\big |}G_{4}{\big |}+{\big |}G_{6}{\big |}+{\big |}G_{7}{\big |}-{\big |}G_{9}{\big |}-{\big |}G_{10}{\big |}+{\big |}G_{12}{\big |}+\cdots =1-{\frac {\sqrt {3}}{\pi }}.\end{aligned}}}$

Candelperger, Coppo[23][24] and Young[7] showed that

${\displaystyle \sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}\cdot H_{n}}{n}}={\frac {\pi ^{2}}{6}}-1,}$

where H_n are the harmonic numbers. Blagouchine[17][25][26][27] provides the following identities

${\displaystyle {\begin{aligned}&\sum _{n=1}^{\infty }{\frac {G_{n}}{n}}=\operatorname {li} (2)-\gamma \\[2mm]&\sum _{n=3}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n-2}}=-{\frac {1}{8}}+{\frac {\ln 2\pi }{12}}-{\frac {\zeta '(2)}{\,2\pi ^{2}}}\\[2mm]&\sum _{n=4}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n-3}}=-{\frac {1}{16}}+{\frac {\ln 2\pi }{24}}-{\frac {\zeta '(2)}{4\pi ^{2}}}+{\frac {\zeta (3)}{8\pi ^{2}}}\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n+2}}={\frac {1}{2}}-2\ln 2+\ln 3\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n+3}}={\frac {1}{3}}-5\ln 2+3\ln 3\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n+k}}={\frac {1}{k}}+\sum _{m=1}^{k}(-1)^{m}{\binom {k}{m}}\ln(m+1)\,,\qquad k=1,2,3,\ldots \\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}}{n^{2}}}=\int _{0}^{1}{\frac {-\operatorname {li} (1-x)+\gamma +\ln x}{x}}\,dx\\[2mm]&\sum _{n=1}^{\infty }{\frac {G_{n}}{n^{2}}}=\int _{0}^{1}{\frac {\operatorname {li} (1+x)-\gamma -\ln x}{x}}\,dx,\end{aligned}}}$

where li(z) is the integral logarithm and ${\displaystyle {\tbinom {k}{m}}}$ 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]

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[18] consider

${\displaystyle \left({\frac {\ln(1+z)}{z}}\right)^{s}=s\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}K_{n}^{(s)}\,,\qquad |z|<1\,,}$

and hence

${\displaystyle G_{n}=-{\frac {K_{n}^{(-1)}}{n!}}}$

Equivalent generalizations were later proposed by Kowalenko[9] and Rubinstein.[30] In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers

${\displaystyle \left({\frac {t}{e^{t}-1}}\right)^{s}=\sum _{k=0}^{\infty }{\frac {t^{k}}{k!}}B_{k}^{(s)},\qquad |t|<2\pi \,,}$

see,[18][28] so that

${\displaystyle G_{n}=-{\frac {B_{n}^{(n-1)}}{(n-1)\,n!}}}$

Jordan[1][16][31] defines polynomials ψ_n(s) such that

${\displaystyle {\frac {z(1+z)^{s}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(s)\,,\qquad |z|<1\,,}$

and call them Bernoulli polynomials of the second kind. From the above, it is clear that G_n = ψ_n(0). Carlitz[16] generalized Jordan's polynomials ψ_n(s) by introducing polynomials β

${\displaystyle \left({\frac {z}{\ln(1+z)}}\right)^{s}\!\!\cdot (1+z)^{x}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\,\beta _{n}^{(s)}(x)\,,\qquad |z|<1\,,}$

and therefore

${\displaystyle G_{n}={\frac {\beta _{n}^{(1)}(0)}{n!}}}$

Blagouchine[17][32] introduced numbers G_n(k) such that

${\displaystyle G_{n}(k)={\frac {1}{n!}}\sum _{\ell =1}^{n}{\frac {s(n,\ell )}{\ell +k}},}$

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[31]

${\displaystyle c_{n}^{(k)}=\sum _{\ell =0}^{n}{\frac {s(n,\ell )}{(\ell +1)^{k}}},}$

so that G_n = c_n⁽¹⁾/n! Numbers c_n^(k) are called by the author poly-Cauchy numbers.[31] Coffey[21] defines polynomials

${\displaystyle P_{n+1}(y)={\frac {1}{n!}}\int _{0}^{y}x(1-x)(2-x)\cdots (n-1-x)\,dx}$

and therefore |G_n| = P_n+1(1).

See also

• Stirling polynomials
• Bernoulli polynomials of the second kind

References

1. Ch. Jordan. The Calculus of Finite Differences Chelsea Publishing Company, USA, 1947.
2. L. Comtet. Advanced combinatorics (2nd Edn.) D. Reidel Publishing Company, Boston, USA, 1974.
3. H.T. Davis. The approximation of logarithmic numbers. Amer. Math. Monthly, vol. 64, no. 8, pp. 11–18, 1957.
4. P. C. Stamper. Table of Gregory coefficients. Math. Comp. vol. 20, p. 465, 1966.
5. D. Merlini, R. Sprugnoli, M. C. Verri. The Cauchy numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006.
6. G. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol. 14, 11.4.8, 2011
7. 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.
8. V. Kowalenko. Properties and Applications of the Reciprocal Logarithm Numbers. Acta Appl. Math., vol. 109, pp. 413–437, 2010.
9. 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.
10. I. M. Alabdulmohsin. Summability calculus, arXiv:1209.5739, 2012.
11. I. M. Alabdulmohsin. Summability calculus: a Comprehensive Theory of Fractional Finite Sums, Springer International Publishing, 2018.
12. 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.
13. Weisstein, Eric W. "Logarithmic Number." From MathWorld—A Wolfram Web Resource.
14. J. C. Kluyver. Euler's constant and natural numbers. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924.
15. J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, New York, USA, 1950.
16. L. Carlitz. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math., vol. 25, pp. 323–330,1961.
17. 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.
18. S.C. Van Veen. Asymptotic expansion of the generalized Bernoulli numbers B_n^(n − 1) for large values of n (n integer). Indag. Math. (Proc.), vol. 13, pp. 335–341, 1951.
19. 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
20. Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)
21. M.W. Coffey. Series representations for the Stieltjes constants. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014.
22. 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.
23. 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.
24. 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
25. OEIS: A269330
26. OEIS: A270857
27. OEIS: A270859
28. N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.
29. 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.
30. M. O. Rubinstein. Identities for the Riemann zeta function Ramanujan J., vol. 27, pp. 29–42, 2012.
31. Takao Komatsu. On poly-Cauchy numbers and polynomials, 2012.
32. 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