Gregory coefficients

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

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

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

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

or the integral

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

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

It implies the finite summation formula

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

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

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

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

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 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 with

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

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

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

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]

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

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

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

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

see,[18][28] so that

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

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

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

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

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

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]

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

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).

• Stirling polynomials
• Bernoulli polynomials of the second kind