Written: May 18, 2019
Dear Doron,
I have (only now ...) seen your remarks on an article by Allouche. The article is very instructive and I agree that it is important to write such a response.
However, I disagree with one point you make.
You write that Allouche should have added
"... and mostly obsolete (thanks to the Wilf-Zeilberger [PWZ] algorithmic proof theory, implemented in Maple, Mathematica and other systems)".
While I agree with a part of it (as I explain below), I regard this otherwise as a truly harmful message.
Yes, when it comes to *proving* a conjectured identity, then it is indeed (usually) not necessary anymore to look into the literature: your algorithm will do it in a fraction of a second (apart from exceptional cases where our computers are powerful enough to finish the computation; sure, this only occurs in very, very rare cases, when there are too many parameters involved).
However, the above situation is not the only situation that one could encounter: it may be that you obtained some sum-expression for your problem, you want to prove that this expression has a certain property, but from the expression that you obtained you don't see that property (say, the expression also cannot be evaluated in closed form). In such a situation, the algorithms can't do anything. And yet something can be done: you apply a (hypergeometric) transformation formula, and it may happen that, from this new expression, you can read off your desired property.
So, I do agree that Allouche is to be critized for his statement "The literature about sums involving binomial coefficients is huge" in the present context. However, the criticism should be, first, that he should mention that binomial sums are (usually) hypergeometric series, and thus the hypergeometric literature should be consulted, and, second, that when it comes to the verification of conjectured identities, the most direct and fastest way to do it is by applying your algorithm.
I am saying all this because it can be observed that the message from above that you send is received, and then, when people encounter a sum that can't be evaluated in closed form, they don't know what to do, respectively believe that nothing can be done, even in situations when (say) a simple, classical 2F1-transformation formula would do the job.
With best wishes,
Christian Krattenthaler