Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm
z ln ( 1 + z ) = 1 + 1 2 z − 1 12 z 2 + 1 24 z 3 − 19 720 z 4 + 3 160 z 5 − 863 60480 z 6 + ⋯ = 1 + ∑ n = 1 ∞ G n z n , | z | < 1 . {\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 Gn = (−1)n−1|Gn| 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.
Numerical values
Computation and representations The simplest way to compute Gregory coefficients is to use the recurrence formula
| G n | = − ∑ k = 1 n − 1 | G k | n + 1 − k + 1 n + 1 {\displaystyle |G_{n}|=-\sum _{k=1}^{n-1}{\frac {|G_{k}|}{n+1-k}}+{\frac {1}{n+1}}}
with G1 = 1/2. Gregory coefficients may be also computed explicitly via the following differential
n ! G n = [ d n d z n z ln ( 1 + z ) ] z = 0 , {\displaystyle 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 = 1 n ! ∫ 0 1 x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) d x = ∫ 0 1 ( x n ) d x , {\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 be proved by integrating ( 1 + z ) x {\displaystyle (1+z)^{x}} between 0 and 1 with respect to x {\displaystyle x} , once directly and the second time using the binomial series expansion first. It implies the finite summation formula
n ! G n = ∑ ℓ = 0 n s ( n , ℓ ) ℓ + 1 , {\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
G n = ( − 1 ) n − 1 ∫ 0 ∞ d x ( 1 + x ) n ( ln 2 x + π 2 ) , {\displaystyle 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
1 6 n ( n − 1 ) < | G n | < 1 6 n , n > 2 , {\displaystyle {\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. In particular,
1 n ln 2 n − 2 n ln 3 n ⩽ | G n | ⩽ 1 n ln 2 n − 2 γ n ln 3 n , n ⩾ 5 . {\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
| G n | ∼ 1 n ln 2 n , n → ∞ . {\displaystyle {\big |}G_{n}{\big |}\sim {\frac {1}{n\ln ^{2}n}},\qquad n\to \infty .}
More accurate description of Gn at large n may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine.
Series with Gregory coefficients Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include
∑ n = 1 ∞ | G n | = 1 ∑ n = 1 ∞ G n = 1 ln 2 − 1 ∑ n = 1 ∞ | G n | n = γ , {\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. More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin and some other authors calculated
∑ n = 2 ∞ | G n | n − 1 = − 1 2 + ln 2 π 2 − γ 2 ∑ n = 1 ∞ | G n | n + 1 = 1 − ln 2. {\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 also gives these identities with
∑ n = 0 ∞ ( − 1 ) n ( | G 3 n + 1 | + | G 3 n + 2 | ) = 3 π ∑ n = 0 ∞ ( − 1 ) n ( | G 3 n + 2 | + | G 3 n + 3 | ) = 2 3 π − 1 ∑ n = 0 ∞ ( − 1 ) n ( | G 3 n + 3 | + | G 3 n + 4 | ) = 1 2 − 3 π . {\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 and Young showed that
∑ n = 1 ∞ | G n | ⋅ H n n = π 2 6 − 1 , {\displaystyle \sum _{n=1}^{\infty }{\frac {{\big |}G_{n}{\big |}\cdot H_{n}}{n}}={\frac {\pi ^{2}}{6}}-1,}
where Hn are the harmonic numbers. Blagouchine provides the following identities
∑ n = 1 ∞ G n n = li ( 2 ) − γ ∑ n = 3 ∞ | G n | n − 2 = − 1 8 + ln 2 π 12 − ζ ′ ( 2 ) 2 π 2 ∑ n = 4 ∞ | G n | n − 3 = − 1 16 + ln 2 π 24 − ζ ′ ( 2 ) 4 π 2 + ζ ( 3 ) 8 π 2 ∑ n = 1 ∞ | G n | n + 2 = 1 2 − 2 ln 2 + ln 3 ∑ n = 1 ∞ | G n | n + 3 = 1 3 − 5 ln 2 + 3 ln 3 ∑ n = 1 ∞ | G n | n + k = 1 k + ∑ m = 1 k ( − 1 ) m ( k m ) ln ( m + 1 ) , k = 1 , 2 , 3 , … ∑ n = 1 ∞ | G n | n 2 = ∫ 0 1 − li ( 1 − x ) + γ + ln x x d x ∑ n = 1 ∞ G n n 2 = ∫ 0 1 li ( 1 + x ) − γ − ln x x d x , {\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 ( k m ) {\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.
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 considered
( ln ( 1 + z ) z ) s = s