Application of Padé approximation to Euler's constant and Stirling's formula (Q829868)

The authors introduce the digamma function as usually by \[\Psi(z)=\Gamma'(z)/\Gamma(z),\ z\in\mathbb{C}\setminus \mathbb{Z}_{\leq 0}=-\gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),\] with Euler's constant defined as \[\gamma=\lim_{k\rightarrow\infty}\,(H_k-\log{(k)}),\] where \(h_k=\sum_{j=1}^k\,1/j\). Further using \[\Phi(z)=\sum_n=0,(-1)^n\,\frac{B_{2n+2}}{2n+2}\,z^n\in\mathbb{Q}[[z]],\] they formulate the first main result \textbf{Theorem 1} For any real number \(x>0\) and any integers \(k\geq 1,n\geq 0\), we have \[\Psi(x)=\log{(x_n)}-\sum_{j=0}^n\,\frac{1}{j+x}+\frac{1}{2x_n}-\frac{1}{x_n^2}[k-1/k]_{\Phi}\left(-\frac{1}{x_n^2}\right)+\varepsilon_k(x_n),\] where \(x_n=x+n\) and \[|\varepsilon_k(x_n)|\leq\frac{(2k+1)(2k+2)}{(4k+3)x_n^{4k+2}}\cdot\frac{(2k)!^2}{\left(\begin{matrix}4k+2\\ 2k+1\end{matrix}\right)^2}.\] Here The \(B_n\) are the Bernoulli numbers and \([k-1/k]_{\Phi}\in\mathbb{Q}(z)\) denotes the Padé approximant of \(\Phi\) with numerator and denominator of degree \(\leq k-1\) and \(\leq k\) respectively. After discussing the flexibility of this result in finding new sequences of approximations of values of \(\Psi(x)\) by choosing \(k\) as a sub-linear function of \(n\), they formulate their second main result by using a Padé approximant of \[\Omega(z)=\sum_{n=0}^{\infty}\,(-1)^{n+1}\,\frac{B_{2n+4}}{(2n+3()2n+4)}\,z^n\in\mathbb{Q}[[z]]\] in \textbf{Theorem 2} For any real number \(x>0\) and any integers \(k\geq 1,n\geq 0\), we have \[\log{(\Gamma(x_n))}=x_n\log{(x_n)}-x_n+\frac{1}{2}\log{(2\pi /x_n)}+\frac{1}{12x_n}-\frac{1}{x_n^3}[k-1/k]_{\Omega}\left(-\frac{1}{x_n^2}\right)+\hat{\varepsilon}(x_n),\] where \(x_n=x+n\) and \[|\hat{\varepsilon}(x_n)|\leq \frac{(2k+1)(2k+2)}{(4k+3)x_n^{4k+2}}\cdot\frac{(2k)!^2}{\left(\begin{matrix}4k+2\\ 2k+1\end{matrix}\right)^2}.\] The paper starts with an introduction (\S1, in which both results and some corollaries are formulated), followed by background material on the concepts and functions used (\S2, \S3). After that the proof of the results theorems 1 and 2 is given in \S4 and \S5.\\ \S6 discusses rational approximations to Euler's constant and in \S7 they look into the connection with Apéry sequences (used in the study of \(\zeta(2)\) and \(\zeta(3)\)); the paper concludes with a list of 14 references. An interesting, well written paper.