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

{\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 $G n = (-1) n -1 | G n |$ 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]

Numerical values

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$...$	OEIS sequences
$G n$	$+.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/2⁠$	$- ⁠ 1 / 12 ⁠$	$+ ⁠ 1 / 24 ⁠$	$- ⁠ 19 / 720 ⁠$	$+ ⁠ 3 / 160 ⁠$	$- ⁠ 863 / 60480 ⁠$	$+ ⁠ 275 / 24192 ⁠$	$- ⁠ 33953 / 3628800 ⁠$	$+ ⁠ 8183 / 1036800 ⁠$	$- ⁠ 3250433 / 479001600 ⁠$	$+ ⁠ 4671 / 788480 ⁠$	$...$	OEIS: A002206 (numerators), OEIS: A002207 (denominators)

Computation and representations

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

|G_{n}|=-\sum _{k=1}^{n-1}{\frac {|G_{k}|}{n+1-k}}+{\frac {1}{n+1}}

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

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

or the integral

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,

which can be proved by integrating $(1+z)^{x}$ between 0 and 1 with respect to $x$ , once directly and the second time using the binomial series expansion first.

It implies the finite summation formula

n!G_{n}=\sum _{\ell =0}^{n}{\frac {s(n,\ell )}{\ell +1}},

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

and Schröder's integral formula^[19]^[20]

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

Bounds and asymptotic behavior

The Gregory coefficients satisfy the bounds

{\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,

{\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]

{\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

{\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

{\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 with

{\begin{aligned}&\sum _{n=0}^{\infty }(-1)^{n}({\big |}G_{3n+1}{\big |}+{\big |}G_{3n+2}{\big |})={\frac {\sqrt {3}}{\pi }}\\[2mm]&\sum _{n=0}^{\infty }(-1)^{n}({\big |}G_{3n+2}{\big |}+{\big |}G_{3n+3}{\big |})={\frac {2{\sqrt {3}}}{\pi }}-1\\[2mm]&\sum _{n=0}^{\infty }(-1)^{n}({\big |}G_{3n+3}{\big |}+{\big |}G_{3n+4}{\big |})={\frac {1}{2}}-{\frac {\sqrt {3}}{\pi }}.\end{aligned}}

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

\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

{\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 ${\tbinom {k}{m}}$ 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

\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

n!G_{n}=-K_{n}^{(-1)}

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

\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

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

Jordan^[1]^[16]^[31] defines polynomials $ψ n (s)$ such that

{\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 $β$

\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

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

Blagouchine^[17]^[32] introduced numbers $G n (k)$ such that

n!G_{n}(k)=\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]

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

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

and therefore $| G n | = P n +1 (1)$ .

References

^ ^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 Alabdulmohsin, Ibrahim M. (2012). "Summability Calculus". arXiv:1209.5739 [math.CA].
^ ^a ^b ^c Alabdulmohsin, Ibrahim M. (2018). Summability Calculus. doi:10.1007/978-3-319-74648-7. ISBN 978-3-319-74647-0.
^ 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, New York, 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 B_n^(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.
^ Blagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009.
^ Candelpergher, Bernard; Coppo, Marc-Antoine (2012). "A new class of identities involving Cauchy numbers, harmonic numbers and zeta values". The Ramanujan Journal. 27 (3): 305–328. doi:10.1007/s11139-011-9361-7.
^ 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.
^ Rubinstein, Michael O. (2012). "Identities for the Riemann zeta function". The Ramanujan Journal. 27: 29–42. doi:10.1007/s11139-010-9276-8.
^ ^a ^b ^c "Takao Komatsu. On poly-Cauchy numbers and polynomials, 2012" (PDF). Archived from the original (PDF) on 2016-03-16. Retrieved 2016-05-20.
^ 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

[jrdn-1] Ch. Jordan. The Calculus of Finite Differences Chelsea Publishing Company, USA, 1947.

[cmt1-2] L. Comtet. Advanced combinatorics (2nd Edn.) D. Reidel Publishing Company, Boston, USA, 1974.

[davis-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.

[merl-5] D. Merlini, R. Sprugnoli, M. C. Verri. The Cauchy numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006.

[nemes-6] G. Nemes. An asymptotic expansion for the Bernoulli numbers of the second kind. J. Integer Seq, vol. 14, 11.4.8, 2011

[young-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.

[kowa1-8] V. Kowalenko. Properties and Applications of the Reciprocal Logarithm Numbers. Acta Appl. Math., vol. 109, pp. 413–437, 2010.

[kowa2-9] 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.

[Alab1-10] Alabdulmohsin, Ibrahim M. (2012). "Summability Calculus". arXiv:1209.5739 [math.CA].

[Alab2-11] Alabdulmohsin, Ibrahim M. (2018). Summability Calculus. doi:10.1007/978-3-319-74648-7. ISBN 978-3-319-74647-0.

[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.

[klvr-14] J. C. Kluyver. Euler's constant and natural numbers. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924.

[stef1-15] J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, New York, USA, 1950.

[car-16] L. Carlitz. A note on Bernoulli and Euler polynomials of the second kind. Scripta Math., vol. 25, pp. 323–330,1961.

[blag1-17] ^ ^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.

[vveen-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.

[blag5-19] 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)

[cfy-21] M.W. Coffey. Series representations for the Stieltjes constants. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014.

[blag2-22] Blagouchine, Iaroslav V. (2015). "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations". Journal of Number Theory. 148: 537–592. arXiv:1401.3724. doi:10.1016/j.jnt.2014.08.009.

[23] Candelpergher, Bernard; Coppo, Marc-Antoine (2012). "A new class of identities involving Cauchy numbers, harmonic numbers and zeta values". The Ramanujan Journal. 27 (3): 305–328. doi:10.1007/s11139-011-9361-7.

[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

[nrd-28] N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.

[blag3-29] 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] Rubinstein, Michael O. (2012). "Identities for the Riemann zeta function". The Ramanujan Journal. 27: 29–42. doi:10.1007/s11139-010-9276-8.

[kmts-31] "Takao Komatsu. On poly-Cauchy numbers and polynomials, 2012" (PDF). Archived from the original (PDF) on 2016-03-16. Retrieved 2016-05-20.

[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

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