Gregory coefficients

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

: $$

|G_n| = -\sum_{k=1}^{n-1} \frac

with {{math|G₁ {{=}} {{sfrac|1|2}}}}.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

: $$

n! G_n=\left[\frac{\textrm d^n}{\textrm dz^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 {{math|s(n,ℓ)}} are the signed Stirling numbers of the first kind.

and Schröder's integral formulaI. 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:1612.03292Ernst Schröder, Zeitschrift fur Mathematik und Physik, vol. 25, pp. 106–117 (1880)

: $$

G_n=(-1)^{n-1} \int_0^\infty \frac{dx}{(1+x)^n(\ln^2 x + \pi^2)},

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. These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine.Ia.V. Blagouchine. [https://dx.doi.org/10.1016/j.jmaa.2016.04.032 Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to {{pi}}⁻¹.] J.Math. Anal. Appl., 2015. 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\geqslant5\,.

Asymptotically, at large index {{math|n}}, these numbers behave as

:$$

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

More accurate description of {{math|G_n}} at large {{math|n}} may be found in works of Van Veen,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. Davis,H.T. Davis. [https://www.jstor.org/stable/2308510 The approximation of logarithmic numbers.] Amer. Math. Monthly, vol. 64, no. 8, pp. 11–18, 1957. Coffey,M.W. Coffey. [https://projecteuclid.org/euclid.rmjm/1407154909 Series representations for the Stieltjes constants]. Rocky Mountain J. Math., vol. 44, pp. 443–477, 2014. NemesG. Nemes. [https://www.kurims.kyoto-u.ac.jp/EMIS/journals/JIS/VOL14/Nemes/nemes4.pdf An asymptotic expansion for the Bernoulli numbers of the second kind]. J. Integer Seq, vol. 14, 11.4.8, 2011 and Blagouchine.

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

:$$

\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}

where {{math|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.{{cite journal | url=http://www.sciencedirect.com/science/article/pii/S0022314X14002820 | doi=10.1016/j.jnt.2014.08.009 | title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations | date=2015 | last1=Blagouchine | first1=Iaroslav V. | journal=Journal of Number Theory | volume=148 | pages=537–592 | arxiv=1401.3724 }} More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin and some other authors calculated

:$$

\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}

Alabdulmohsin{{cite arXiv | eprint=1209.5739 | last1=Alabdulmohsin | first1=Ibrahim M. | title=Summability Calculus | date=2012 | class=math.CA }}{{cite book | url=https://link.springer.com/book/10.1007/978-3-319-74648-7 | doi=10.1007/978-3-319-74648-7 | title=Summability Calculus | date=2018 | last1=Alabdulmohsin | first1=Ibrahim M. | isbn=978-3-319-74647-0 }} also gives these identities with

:$$

\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}

Candelperger, Coppo{{cite journal | url=https://link.springer.com/article/10.1007%2Fs11139-011-9361-7 | doi=10.1007/s11139-011-9361-7 | title=A new class of identities involving Cauchy numbers, harmonic numbers and zeta values | date=2012 | last1=Candelpergher | first1=Bernard | last2=Coppo | first2=Marc-Antoine | journal=The Ramanujan Journal | volume=27 | issue=3 | pages=305–328 }}B. Candelpergher and M.-A. Coppo. [https://hal.inria.fr/file/index/docid/634313/filename/ArtZetamod.pdf 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

:$$

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

where {{math|H_n}} are the harmonic numbers.

Blagouchine{{OEIS2C|id=A269330}}{{OEIS2C|id=A270857}}{{OEIS2C|id=A270859}} provides the following identities

:$$

\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}

where {{math|li(z)}} is the integral logarithm and $\backslash tbinom\{k\}\{m\}$ 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.N. Nörlund. Vorlesungen über Differenzenrechnung. Springer, Berlin, 1924.Ia.V. Blagouchine. [http://www.sciencedirect.com/science/article/pii/S0022314X15002255 Expansions of generalized Euler's constants into the series of polynomials in {{pi}}⁻² and into the formal enveloping series with rational coefficients only.] J. Number Theory, vol. 158, pp. 365–396, 2016.

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

:$$

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

and hence

: $$

n!G_n=-K_n^{(-1)}

Equivalent generalizations were later proposed by Kowalenko and Rubinstein.{{cite journal | url=https://link.springer.com/article/10.1007%2Fs11139-010-9276-8 | doi=10.1007/s11139-010-9276-8 | title=Identities for the Riemann zeta function | date=2012 | last1=Rubinstein | first1=Michael O. | journal=The Ramanujan Journal | volume=27 | pages=29–42 | arxiv=0812.2592 }} 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^{(s)}_k , \qquad |t|<2\pi\,,

see, so that

: $$

n!G_n=-\frac{B_n^{(n-1)}}{n-1}

Jordan{{Cite web |url=https://carma.newcastle.edu.au/alfcon/pdfs/Takao_Komatsu-alfcon.pdf |title=Takao Komatsu. On poly-Cauchy numbers and polynomials, 2012. |access-date=2016-05-20 |archive-date=2016-03-16 |archive-url=https://web.archive.org/web/20160316055106/https://www.carma.newcastle.edu.au/alfcon/pdfs/Takao_Komatsu-alfcon.pdf |url-status=dead }} defines polynomials {{math|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 {{math|G_n {{=}} ψ_n(0)}}.

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 {{math|ψ_n(s)}} by introducing polynomials {{math|β}}

:$$

\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\,,

and therefore

: $$

n!G_n=\beta^{(1)}_n(0)

Blagouchine

Ia.V. Blagouchine. [http://math.colgate.edu/~integers/sjs3/sjs3.pdf 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 {{math|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 {{math|n}}. Clearly, {{math|G_n {{=}} G_n(1)}}. These numbers are strictly alternating {{math|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

: $$

c_n^{(k)}=\sum_{\ell=0}^n \frac{s(n,\ell)}{(\ell+1)^k},

so that {{math|G_n {{=}} c_n⁽¹⁾/n!}} Numbers {{math|c_n^(k)}} are called by the author poly-Cauchy numbers. Coffey

defines polynomials

: $$

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

and therefore {{math|{{!}}G_n{{!}} {{=}} P_n+1(1)}}.

• Stirling polynomials
• Bernoulli polynomials of the second kind

{{Reflist|2}}

Category:Integer sequences

Category:Number theory

Category:Inequalities (mathematics)

Category:Asymptotic analysis