View source for Gregory coefficients

'''Gregory coefficients''' {{math|''G''<sub>''n''</sub>}}, also known as '''reciprocal logarithmic numbers''', '''Bernoulli numbers of the second kind''', or '''Cauchy numbers of the first kind''',<ref name="jrdn">Ch. Jordan. ''The Calculus of Finite Differences'' Chelsea Publishing Company, USA, 1947.</ref><ref name="cmt1">L. Comtet. ''Advanced combinatorics (2nd Edn.)'' D. Reidel Publishing Company, Boston, USA, 1974.</ref><ref name="davis" /><ref>P. C. Stamper. ''Table of Gregory coefficients''. Math. Comp. vol. 20, p. 465, 1966.</ref><ref name="merl">[http://www.sciencedirect.com/science/article/pii/S0012365X06002512 D. Merlini, R. Sprugnoli, M. C. Verri. ''The Cauchy numbers''. Discrete Math., vol. 306, pp. 1906–1920, 2006.]</ref><ref name="nemes" /><ref name="young">[http://www.sciencedirect.com/science/article/pii/S0022314X0800067X 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.]</ref><ref name="kowa1">[https://link.springer.com/article/10.1007%2Fs10440-008-9325-0 V. Kowalenko. ''Properties and Applications of the Reciprocal Logarithm Numbers''. Acta Appl. Math., vol. 109, pp. 413–437, 2010.]</ref><ref name="kowa2">[https://link.springer.com/article/10.1007%2Fs10440-008-9304-5 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.]</ref><ref name="Alab1" /><ref name="Alab2" /><ref>[http://www.mathnet.or.kr/mathnet/thesis_file/BKMS-52-3-987-998.pdf 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.]</ref><ref>[http://mathworld.wolfram.com/LogarithmicNumber.html Weisstein, Eric W. "Logarithmic Number." From MathWorld—A Wolfram Web Resource.]</ref> are the rational numbers 
that occur in the [[Maclaurin series]] expansion of the reciprocal logarithm

:<math>
\begin{align}
\frac{z}{\ln(1+z)} & = 1+\frac12 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{align}
</math>

Gregory coefficients are alternating {{math|''G''<sub>''n''</sub> {{=}} (−1)<sup>''n''−1</sup>{{!}}''G''<sub>''n''</sub>{{!}}}} for {{math|''n'' > 0
}} and decreasing in absolute value. These numbers are named after [[Biography:James Gregory (mathematician)|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.<ref name="jrdn" /><ref name="merl" /><ref name="klvr" /><ref name="stef1">J.F. Steffensen. ''Interpolation (2nd Edn.).'' Chelsea Publishing Company, New York, USA, 1950.</ref><ref name="car" /><ref name="blag1" />

==Numerical values==

{| class="wikitable" border="1" align="center"
|-
! {{math|''n''}}
! width="40" |{{math|1}}
! width="40" |{{math|2}}
! width="50" |{{math|3}}
! width="50" |{{math|4}}
! width="50" |{{math|5}}
! width="50" |{{math|6}}
! width="50" |{{math|7}}
! width="50" |{{math|8}}
! width="50" |{{math|9}}
! width="80" |{{math|10}}
! width="80" |{{math|11}}
! width="80" |{{math|...}}
!| OEIS sequences
|- align="center"
! {{math|''G<sub>n</sub>''}}
| {{math|+{{sfrac|1|2}}}}
| {{math|&minus;{{sfrac|1|12}}}}
| {{math|+{{sfrac|1|24}}}}
| {{math|&minus;{{sfrac|19|720}}}}
| {{math|+{{sfrac|3|160}}}}
| {{math|&minus;{{sfrac|863|60480}}}}
| {{math|+{{sfrac|275|24192}}}}
| {{math|&minus;{{sfrac|33953|3628800}}}}
| {{math|+{{sfrac|8183|1036800}}}}
| {{math|&minus;{{sfrac|3250433|479001600}}}}
| {{math|+{{sfrac|4671|788480}}}}
| {{math|...}}
| {{OEIS2C|id=A002206}} (numerators),
{{OEIS2C|id=A002207}} (denominators)
|}

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

: <math>
|G_n| = -\sum_{k=1}^{n-1} \frac{|G_k|}{n+1-k} + \frac 1 {n+1}
</math>

with {{math|''G''<sub>1</sub> {{=}} {{sfrac|1|2}}}}.<ref name="klvr">J. C. Kluyver. ''Euler's constant and natural numbers''. Proc. K. Ned. Akad. Wet., vol. 27(1-2), 1924.</ref><ref name="vveen" /> Gregory coefficients may be also computed explicitly via the following differential

: <math>
n! G_n=\left[\frac{\textrm d^n}{\textrm dz^n}\frac{z}{\ln(1+z)}\right]_{z=0},
</math>

or the integral

: <math>
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,
</math>
which can be proved by integrating <math> (1+z)^x </math>  between 0 and 1 with respect to <math> x </math>, once directly and the second time using the [[Binomial series|binomial series]] expansion first.

It implies the finite summation formula
: <math>
n! G_n= \sum_{\ell=0}^n \frac{s(n,\ell)}{\ell+1} , 
</math>
where {{math|''s''(''n'',''ℓ'')}} are the signed [[Stirling numbers of the first kind]].

and Schröder's integral formula<ref name="blag5">I. V. Blagouchine, [https://cs.uwaterloo.ca/journals/JIS/VOL20/Blagouchine/blag5.html 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</ref><ref>Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)</ref>
: <math>
G_n=(-1)^{n-1} \int_0^\infty \frac{dx}{(1+x)^n(\ln^2 x + \pi^2)},
</math>

==Bounds and asymptotic behavior==

The Gregory coefficients satisfy the bounds

:<math>
\frac{1}{6n(n-1)}<\big|G_n\big|<\frac{1}{6n},\qquad n>2,
</math>

given by Johan Steffensen.<ref name="stef1" /> These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine.<ref name="blag1">
[https://dx.doi.org/10.1016/j.jmaa.2016.04.032 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 {{pi}}<sup>&minus;1</sup>.'' J.Math. Anal. Appl., 2015.]</ref> In particular,

:<math>
\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\geqslant5\,.
</math>

Asymptotically, at large index {{math|''n''}}, these numbers behave as<ref name="cmt1" /><ref name="blag1" /><ref name="blag5" />

:<math>
\big|G_n\big|\sim \frac{1}{n\ln^2 n}, \qquad n\to\infty.
</math>

More accurate description of {{math|''G''<sub>''n''</sub>}} at large {{math|''n''}} may be found in works of Van Veen,<ref name="vveen">S.C. Van Veen. ''Asymptotic expansion of the generalized Bernoulli numbers'' ''B''<sub>''n''</sub><sup>(''n''&nbsp;−&nbsp;1)</sup> ''for large values of ''n'' (''n'' integer)''. Indag. Math. (Proc.), vol. 13, pp. 335–341, 1951.</ref> Davis,<ref name="davis">[https://www.jstor.org/stable/2308510 H.T. Davis. ''The approximation of logarithmic numbers''. Amer. Math. Monthly, vol. 64, no. 8, pp. 11–18, 1957.]</ref> Coffey,<ref name="cfy">[https://projecteuclid.org/euclid.rmjm/1407154909 M.W. Coffey. ''Series representations for the Stieltjes constants''. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014.]</ref> Nemes<ref name="nemes">G. Nemes. ''An asymptotic expansion for the Bernoulli numbers of the second kind''. J. Integer Seq, vol. 14, 11.4.8, 2011</ref> and Blagouchine.<ref name="blag1" />

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

:<math>
\begin{align}
&\sum_{n=1}^\infty\big|G_n\big|=1 \\[2mm]
&\sum_{n=1}^\infty G_n=\frac{1}{\ln2} -1 \\[2mm]
&\sum_{n=1}^\infty \frac{\big|G_n\big|}{n}=\gamma,
\end{align}
</math>

where {{math|''&gamma;'' {{=}} 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 [[Biography:Lorenzo Mascheroni|Lorenzo Mascheroni]].<ref name="blag1" /><ref name="blag2">
[http://www.sciencedirect.com/science/article/pii/S0022314X14002820 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.]</ref> More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko,<ref name="kowa1" /> Alabdulmohsin <ref name="Alab1" /><ref name="Alab2" /> and some other authors calculated

:<math>
\begin{array}{l}
\displaystyle
\sum_{n=2}^\infty \frac{\big|G_n\big|}{n-1}= -\frac{1}{2} + \frac{\ln2\pi}{2} -\frac{\gamma}{2}  \\[6mm]
\displaystyle
\displaystyle\sum_{n=1}^{\infty}\!\frac{\big|G_n\big|}{n+1}= 1- \ln2.
\end{array}
</math>

Alabdulmohsin<ref name="Alab1">[https://arxiv.org/abs/1209.5739 I. M. Alabdulmohsin. ''Summability calculus'', arXiv:1209.5739, 2012.]</ref><ref name="Alab2">[https://www.springer.com/gp/book/9783319746470 I. M. Alabdulmohsin. ''Summability calculus: a Comprehensive Theory of Fractional Finite Sums'', Springer International Publishing, 2018.]</ref> also gives these identities with

:<math>
\begin{align}
& \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{align} 
</math>

Candelperger, Coppo<ref>
[https://link.springer.com/article/10.1007%2Fs11139-011-9361-7 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.]</ref><ref>[https://hal.inria.fr/file/index/docid/634313/filename/ArtZetamod.pdf 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]</ref> and Young<ref name="young" /> showed that

:<math>
\sum_{n=1}^\infty \frac{\big|G_n\big|\cdot H_n}{n}=\frac{\pi^2}{6}-1, 
</math>

where {{math|''H''<sub>''n''</sub>}} are the [[Harmonic number|harmonic number]]s.
Blagouchine<ref name="blag1" /><ref>{{OEIS2C|id=A269330}}</ref><ref>{{OEIS2C|id=A270857}}</ref><ref>{{OEIS2C|id=A270859}}</ref> provides the following identities

:<math>
\begin{align}
& \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{\ln2\pi}{12} - \frac{\zeta'(2)}{\,2\pi^2}\\[2mm]
& \sum_{n=4}^\infty \frac{\big|G_n\big|}{n-3} =
-\frac{1}{16} + \frac{\ln2\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\ln2 +\ln3  \\[2mm] 
& \sum_{n=1}^\infty \frac{\big|G_n\big|}{n+3} =\frac{1}{3}-5\ln2+3\ln3  \\[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{align}
</math>

where {{math|li(''z'')}} is the [[Logarithmic integral function|integral logarithm]] and <math>\tbinom{k}{m}</math> is the [[Binomial coefficient|binomial coefficient]].
It is also known that the [[Riemann zeta function|zeta function]], the [[Gamma function|gamma function]], the [[Polygamma function|polygamma function]]s, the [[Stieltjes constants]] and many other special functions and constants may be expressed in terms of infinite series containing these numbers.<ref name="jrdn" /><ref name="blag1" /><ref name="vveen" /><ref name="nrd">N. Nörlund. ''Vorlesungen über Differenzenrechnung''. Springer, Berlin, 1924.</ref><ref name="blag3">[http://www.sciencedirect.com/science/article/pii/S0022314X15002255 Ia.V. Blagouchine. ''Expansions of generalized Euler's constants into the series of polynomials in {{pi}}<sup>&minus;2</sup> and into the formal enveloping series with rational coefficients only'' J. Number Theory, vol. 158, pp. 365–396, 2016.]</ref>

==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<ref name="vveen" /> consider

:<math>
\left(\frac{\ln(1+z)}{z}\right)^s= s\sum_{n=0}^\infty \frac{z^n}{n!}K^{(s)}_n \,,\qquad |z|<1\,,
</math>

and hence

: <math>
n!G_n=-K_n^{(-1)}
</math>

Equivalent generalizations were later proposed by Kowalenko<ref name="kowa2" /> and Rubinstein.<ref>[https://link.springer.com/article/10.1007%2Fs11139-010-9276-8 M. O. Rubinstein. ''Identities for the Riemann zeta function'' Ramanujan J., vol. 27, pp. 29–42, 2012.]</ref> In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers

: <math>
\left(\frac{t}{e^t-1}\right)^s= \sum_{k=0}^\infty \frac{t^k}{k!} B^{(s)}_k , \qquad |t|<2\pi\,,
</math>

see,<ref name="vveen" /><ref name="nrd" /> so that

: <math>
n!G_n=-\frac{B_n^{(n-1)}}{n-1} 
</math>

Jordan<ref name="jrdn" /><ref name="car" /><ref name="kmts">[https://carma.newcastle.edu.au/alfcon/pdfs/Takao_Komatsu-alfcon.pdf Takao Komatsu. ''On poly-Cauchy numbers and polynomials'', 2012.]</ref> defines polynomials {{math|''ψ<sub>n</sub>''(''s'')}} such that

: <math>
\frac{z(1+z)^s}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(s) \,,\qquad |z|<1\,,
</math>

and call them ''[[Bernoulli polynomials of the second kind]]''. From the above, it is clear that {{math|''G''<sub>''n''</sub> {{=}} ''&psi;<sub>n</sub>''(0)}}.
Carlitz<ref name="car">L. Carlitz. ''A note on Bernoulli and Euler polynomials of the second kind''. Scripta Math., vol. 25, pp. 323–330,1961.</ref> generalized Jordan's polynomials {{math|''&psi;''<sub>''n''</sub>(''s'')}} by introducing polynomials {{math|''&beta;''}}

:<math>
\left(\frac{z}{\ln(1+z)}\right)^s \!\!\cdot (1+z)^x= \sum_{n=0}^\infty \frac{z^n}{n!}\,\beta^{(s)}_n(x) \,,\qquad |z|<1\,,
</math>

and therefore

: <math>
n!G_n=\beta^{(1)}_n(0)
</math>

Blagouchine<ref name="blag1" /><ref>
[http://math.colgate.edu/~integers/sjs3/sjs3.pdf 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.] [https://arxiv.org/abs/1606.02044 arXiv:1606.02044]</ref> introduced numbers {{math|''G''<sub>''n''</sub>(''k'')}} such that

: <math>
n!G_n(k)=\sum_{\ell=1}^n \frac{s(n,\ell)}{\ell+k} , 
</math>

obtained their generating function and studied their asymptotics at large {{math|''n''}}. Clearly, {{math|''G''<sub>''n''</sub> {{=}} ''G''<sub>''n''</sub>(1)}}. These numbers are strictly alternating {{math|''G''<sub>''n''</sub>(''k'') {{=}} (-1)<sup>''n''-1</sup>{{!}}''G''<sub>''n''</sub>(''k''){{!}}}} and involved in various expansions for the zeta-functions, [[Euler's constant]] and [[Polygamma function|polygamma function]]s.
A different generalization of the same kind was also proposed by Komatsu<ref name="kmts" />  
: <math>
c_n^{(k)}=\sum_{\ell=0}^n \frac{s(n,\ell)}{(\ell+1)^k},
</math>

so that {{math|''G''<sub>''n''</sub> {{=}} ''c''<sub>''n''</sub><sup>(1)</sup>/''n''!}} Numbers {{math|''c''<sub>''n''</sub><sup>(''k'')</sup>}} are called by the author ''poly-Cauchy numbers''.<ref name="kmts" /> Coffey<ref name="cfy" />
defines polynomials

: <math>
P_{n+1}(y)=\frac 1 {n!} \int_0^y x(1-x)(2-x)\cdots(n-1-x)\, dx 
</math>

and therefore {{math|{{!}}''G''<sub>''n''</sub>{{!}} {{=}} ''P''<sub>''n''+1</sub>(1)}}.

== See also==
* [[Stirling polynomials]]
* [[Bernoulli polynomials of the second kind]]

== References ==
{{Reflist|2}}

[[Category:Integer sequences]]
[[Category:Number theory]]
[[Category:Inequalities]]
[[Category:Asymptotic analysis]]

{{Sourceattribution|Gregory coefficients}}