Padé Approximant Related to Ramanujan’s Formula for the Gamma Function

Ramanujan presented the following approximation to the gamma function: Γ(x+1)∼πxex8x3+4x2+x+1301/6,x→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (x+1)\sim \sqrt{\pi }\left( \frac{x}{e}\right) ^{x} \left( 8x^3+4x^2+x+\frac{1}{30}\right) ^{1/6},\qquad x\rightarrow \infty . \end{aligned}$$\end{document}Based on the Padé approximation method, in this paper we develop Ramanujan’s approximation formula to produce a general result. More precisely, we determine the coefficients aj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_j$$\end{document} and bj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_j$$\end{document} such that Γ(x+1)=πxex8x31+∑j=1pajx-j1+∑j=1qbjx-j+O1xp+q-21/6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (x+1)=\sqrt{\pi }\left( \frac{x}{e}\right) ^{x} \left[ 8x^3\left( \frac{1+\sum _{j=1}^{p}a_jx^{-j}}{1+\sum _{j=1}^{q}b_jx^{-j}} \right) +O\left( \frac{1}{x^{p+q-2}}\right) \right] ^{1/6} \end{aligned}$$\end{document}as x→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow \infty $$\end{document}, where p≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 0$$\end{document} and q≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 0$$\end{document} are any given integers (an empty sum is understood to be zero). In particular, setting (p,q)=(3,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,q)=(3,0)$$\end{document} yields Ramanujan’s approximation to the gamma function. Based on the obtained result, we establish new bounds for the gamma function, improving the double inequality presented by Ramanujan.