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

Abstract

Ramanujan presented the following approximation to the gamma function:

$$\begin{aligned} \Gamma (x+1)\sim \sqrt{\pi }\left( \frac{x}{e}\right) ^{x} \left( 8x^3+4x^2+x+\frac{1}{30}\right) ^{1/6},\qquad x\rightarrow \infty . \end{aligned}$$

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 $a_j$ and $b_j$ such that

$$\begin{aligned} \Gamma (x+1)=\sqrt{\pi }\left( \frac{x}{e}\right) ^{x} \left[ 8x^3\left( \frac{1+\sum _{j=1}^{p}a_jx^{-j}}{1+\sum _{j=1}^{q}b_jx^{-j}} \right) +O\left( \frac{1}{x^{p+q-2}}\right) \right] ^{1/6} \end{aligned}$$

as $x\rightarrow \infty$, where $p\ge 0$ and $q\ge 0$ are any given integers (an empty sum is understood to be zero). In particular, setting $(p,q)=(3,0)$ 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.