Asymptotic expansions of truncated hypergeometric series for $1/\pi$

Abstract

In this paper, we consider rational hypergeometric series of the form \[\frac{p}{\pi}= \sum_{k=0}^\infty u_k\quad\text{with}\quad u_k=\frac{\left(\frac{1}{2}\right)_k \left(q\right)_k \left(1-q\right)_k}{(k!)^3}(r+s\,k)\,t^k,\] where $(a)_k$ denotes the Pochhammer symbol and $p,q,r,s,t$ are algebraic coefficients. Using only the first $n+1$ terms of this series, we define the remainder \[\mathcal{R}_n = \frac{p}{\pi} - \sum_{k=0}^n u_k=\sum_{k=n+1}^\infty u_k.\] We consider an asymptotic expansion of $\mathcal{R}_n$. More precisely, we provide a recursive relation for determining the coefficients $c_j$ such that \[ \mathcal{R}_n = \frac{\left(\frac{1}{2}\right)_n \left(q\right)_n \left(1-q\right)_n}{n!^3}nt^n\left(\sum_{j=0}^{J-1}\frac{c_j}{n^j}+\mathcal{O}\left(n^{-J}\right)\right),\qquad n \rightarrow \infty.\] Here we need $J<\infty$ to approximate $\mathcal{R}_n$, because (like the Stirling series) this series diverges if $J\rightarrow\infty$. By applying our recursive relation to the Chudnovsky formula, we solve an open problem posed by Han and Chen.