Some Remarks on a recent article by J. -P. Allouche

TL;DR

This paper critiques a recent explanation of an identity equivalence, emphasizing that both identities are easily provable with existing algorithms, and questions the validity of the recent reasoning.

Contribution

It clarifies that the recent explanation of the identity equivalence is likely incorrect and highlights the effectiveness of algorithmic proof methods for such identities.

Findings

Both identities are provable using Wilf-Zeilberger algorithms.

The recent explanation by J.-P. Allouche is probably based on a misunderstanding.

Algorithmic proof methods are sufficient for establishing the identities.

Abstract

In 1980 Otto G. Ruehr made some puzzling comments that a certain identity A, that he proved, is equivalent to another identity B, but he did not explain why they are equivalent. Recently J.-P. Allouche tried to explain why they are "equivalent", using about eight pages. We comment that it is extremely unlikely to be Ruehr's reasoning (he probably got mixed up with a different problem), but be that as it may, it is not worthwhile to try and deduce B from A, since both A and B are routinely provable using Wilf-Zeilberger algorithmic proof theory (implemented in Maple) and the Almkvist-Zeilberger algorithm.