Geometrically convergent sequences of upper and lower bounds on the Wallis ratio and related expressions

TL;DR

This paper presents sequences of algebraic upper and lower bounds on the Wallis ratio that converge geometrically to the true value, with implications for computational methods and probability distributions.

Contribution

It introduces new sequences of bounds with geometric convergence for the Wallis ratio and related functions, extending previous results and discussing computational aspects.

Findings

Relative errors converge to 0 geometrically on [x_0, ∞)

Absolute errors tend to 0 as x→∞

Results are applicable to bounds involving the Student distribution

Abstract

Sequences of algebraic upper and lower bounds on the Wallis ratio are given with the relative errors that converge to 0 geometrically and uniformly on any interval of the form [x_0,\infty) for x_0>-\frac12; moreover, the relative and absolute errors converge to 0 as x\to\infty. These conclusions are based on corresponding results for the digamma function \psi:=\Ga'/\Ga. Relations with other relevant results are discussed, as well as the corresponding computational aspects. This work was motivated by studies of exact bounds involving the Student probability distribution.