An iterative approach toward hypergeometric accelerations
John M. Campbell
TL;DR
The paper develops an iterative hypergeometric acceleration framework to generate Ramanujan-type formulas from irregular Pochhammer combinations that resist classical elliptic-function methods. By combining Wilf’s acceleration with Zeilberger’s algorithm and Ω-sums, it derives iteration patterns such as (1,1), unary (1,0), and ternary (1,1,1), yielding closed forms for auxiliary sums and transforming them into rapidly convergent series. The method produces numerous new Ramanujan-type identities for fundamental constants including $\pi$, $\pi^2$, $\sqrt{2}$, and $\log 2$, recovers classical results, and unifies prior results from Ramanujan, Guillera, Chu, and Zhang. It also extends to irregular Ramanujan-type series, nested radicals, and Bauer–Ramanujan-type forms, offering a versatile toolkit for constructing high-precision, fast-converging series with broad applicability.
Abstract
Each of Ramanujan's series for $\frac{1}π$ is of the form $$ \sum_{n=0}^{\infty} z^n \frac{ (a_{1})_{n} (a_{2})_{n} (a_{3})_{n} }{ (b_{1})_{n} (b_{2})_{n} (b_{3})_{n} } (c_{1} n + c_2) $$ for rational parameters such that the difference between the arguments of any lower and upper Pochhammer symbols is not an integer. In accordance with the work of Chu, if an infinite sum of this form admits a closed form, then this provides a formula of Ramanujan type. Chu has introduced remarkable results on formulas of Ramanujan type, through the use of accelerations based on $Ω$-sums related to classical hypergeometric identities. Building on our past work on an acceleration method due to Wilf relying on inhomogeneous difference equations derived from Zeilberger's algorithm, we extend this method through what we refer to as an iterative approach that is inspired by Chu's accelerations derived using iteration patterns for well-poised $Ω$-sums and that we apply to introduce and prove many accelerated formulas of Ramanujan type for universal constants, along with many further accelerations related to the discoveries of Ramanujan, Guillera, and Chu.