Gregory coefficients

==Computation and representations==

The simplest way to compute Gregory coefficients is to use the recurrence formula : with . : 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 : \frac{1}{6n(n-1)}2, given by Johan Steffensen. 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 , these numbers behave as 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 : \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 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, 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 and Young 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 is the integral logarithm and \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 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| see, defines polynomials such that : \frac{z(1+z)^s}{\ln(1+z)}= \sum_{n=0}^\infty z^n \psi_n(s) \,,\qquad |z| and call them Bernoulli polynomials of the second kind. From the above, it is clear that . Carlitz generalized Jordan's polynomials by introducing polynomials : \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| and therefore : n!G_n=\beta^{(1)}_n(0) Blagouchine introduced numbers 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 . Clearly, . These numbers are strictly alternating 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 Numbers 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 . == See also==