On an identity that appears in Apéry's proof of the irrationality of $\zeta(3)$

Question

In proving the irrationality of $\displaystyle \zeta(3)=\sum_{n\geq1}\frac{1}{n^3}$, Apéry introduced the accelerated series

$$ \zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {\binom {2k}{k}k^{3}}}. $$

A crucial ingredient of his proof of the above identity is the well-known equality $$ \sum_{k=1}^{n-1} \frac{(-1)^{k-1}(k-1)!^2}{(n^2-1^2)\ldots(n^2-k^2)}=\frac{1}{n^2}-\frac{2(-1)^{n-1}}{n^2 \binom{2n}{n}}. $$ (note that an excellent overview of his proof can be found in Van der Poorten's "A Proof that Euler Missed...".)

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I was trying to verify the above equality numerically, in Wolfram Mathematica. However, when asking to evaluate the LHS, Mathematica surprisingly gave:

1/n^2

This is obviously false, as quick numerical verification reveals.

This was the simple Mathematica instruction I ran:

Clear[n]
N[Sum[((-1)^(k - 1) ((k - 1)!)^2)/(Product[(n^2 - j^2), {j, 1, k}]), {k, 1, n - 1}]]

Note that Wolfram Alpha gives the same incorrect evaluation for the same expression, as can be seen here.

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So what am I doing wrong? Clearly, the instruction itself is correct (click the Alpha link to see this). So is the Wolfram engine making an elementary mistake, or am I doing so?

Answer 1

You already seem to have indication from Mathematica SE site that this might be a bug, here is I think an interesting observation (too long for a comment).

I reproduced the same issue in another CAS system - Maple 2020. Specifically

simplify(sum((-1)^(k-1)*(k-1)!^2/product(n^2-i^2,i=1..k),k=1..n-1))

gives the result $\frac{1}{n^2}$, which is of course incorrect for positive integer $n$. However it seems the tool somehow makes a sense of the input for non-integral $n$ and keeps going with that...

If we look at what is the intermediate result (before passing it to simplify), it is an expression in terms of gamma function, specifically:

$$ \frac{(1-n)(1+n)\Gamma(1-n)\Gamma(1+n)}{n^2\Gamma(2-n)\Gamma(2+n)}\tag{*}. $$

However for integer $n \geq 1$ it is not hard to see that this expression is not even defined (specifically $\Gamma(1-n)$ and $\Gamma(2-n)$ are not defined because gamma function is not defined for negative integers). So somehow Maple simplified the sum to a result that is more generic, but is not defined for integral values, which paradoxically are the only ones for which the original sum was defined....

Then of course the final simplifcation is done in general without any assumption on $n$, and so for all $n$ for which the $(*)$ is defined, it simplifies to $\frac{1}{n^2}$.

One way around this is to add additional assumptions, in this case we want $n$ to be integer, and indeed if we add assume(n,integer) before the command, we get...

Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

Well it is not what we hoped for, but at least it is giving us an error instead of incorrect result (clearly it is now aware of the limitations of methods in the given domain).

I can only assume it is a similar case in Mathematica...