On two-term hypergeometric recursions with free lower parameters (Q6581932)

In this paper, the authors expand on their previous work [J. Difference Equ. Appl. 29, No. 3, 366--376 (2023; Zbl 1535.33032)] related to series acceleration formulas based on evaluations of hypergeometric functions using Wilf-Zeilberger theory, that pursues the idea of accelerating series using functions that satisfy a linear recurrence with polynomial coefficients derived via Zeilberger's algorithm. This is studied in great detail specifically for functions of the form \[F(n,k) = \frac{(a)_k(b)_k}{(\alpha_1+\beta_1n)_k(\alpha_2+\beta_2n)_k},\] with free parameters \(a\) and \(b\), fixed parameters \(\alpha_1, \beta_1, \alpha_2\), \(\beta_2\) with \(\beta_1,\beta_2\in\mathbb{N}\), and \((x)_k:=x(x+1)\cdots(x+k-1)\) for \(k\in\mathbb{N}\). The method described here is shown to be effective in the sense that it provides a generalized structure for discovering new formulas (e.g., a hypergeometric sum that converge to a nice constant) as well as reaffirming previously discovered well-known series evaluations. A table is provided that summarizes all of their convergence results and the corresponding constants, some of which were found by systematically searching through tuples \((a,b,n)\) with rational entries and instantiating the formulas derived from Zeilberger's algorithm on \(F(n,k)\).\N\NIt is, however, not entirely clear what is the underlying pattern of \((a,b,n)\) that would lend itself naturally to convergence and whether or not these choices influence the convergence rate in a particular way. A generalization of the method described in this paper would necessarily involve analysis away from the form \(F(n,k)\), and may require exploration of functions that satisfy other recurrence equations (possibly a system of such equations).