On two-term hypergeometric recursions with free lower parameters

Abstract

Let $F(n,k)$ be a hypergeometric function that may be expressed so that $n$ appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $\frac{1}{(n)_{k}}$. Only in exceptional cases is $F(n, k)$ such that Zeilberger's algorithm produces a two-term recursion for $\sum_{k = 0}^{\infty} F(n, k)$ obtained via the telescoping of the right-hand side of a difference equation of the form $p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k)$ for fixed $r \in \mathbb{N}$ and polynomials $p_{1}$ and $p_{2}$. Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupa\c{s}, and Amdeberhan are special cases of hypergeometric transforms introduced in our article.