Gregory coefficients

Gregory coefficients G_n, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind and the Cauchy numbers of the first kind,^{[1][2][3][4][5][6][7][8][9][10][11][12][13]} are the rational numbers

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	...	A002206 (numerators),  A002207 (denominators)

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 famous mathematicians and often appear in works of modern authors who do not recognize them.^{[1][5][14][15][16][17]}

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 following integrals

${\displaystyle {\begin{array}{l}\displaystyle G_{n}={\frac {1}{n!}}\int \limits _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx\\[6mm]\displaystyle G_{n}=(-1)^{n-1}\!\int \limits _{0}^{\infty }{\frac {dx}{(1+x)^{n}\left(\ln ^{2}x+\pi ^{2}\right)}},\end{array}}}$

or the finite summation formula

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

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]

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

More accurate description of G_n at large n may be found in works of Van Veen,^[18] Davis,^[3] Coffey,^[19] 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][20] More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko,^[8] Alabdulmohsin ^[10] 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] 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^[21][22] 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][23][24][25]} provides the following identities

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

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][26][27]}

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.^[28] 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][26] so that

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

Jordan^[1][16][29] 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] introduced numbers G_n(k) such that

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

and studied their asymptotics at large n. Clearly, G_n = G_n(1). A different generalization of the same kind was also proposed by Komatsu^[29]

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

so that G_n = c_n⁽¹⁾/n! Numbers c_n^(k) are called by the author poly-Cauchy numbers.^[29] Coffey^[19] defines polynomials

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

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

References

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