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Background

In the following question and references therein, a number of "Sum equals integral" identities are described.

For instance, we have $$ \sum_{n = -\infty}^{+\infty} {\rm sinc} (x)^{N}\, = \int_{-\infty}^{+\infty} {\rm sinc} (x)^{N}\, dx = \pi, \tag{1}\label{1}$$

for $1\leq N \leq 6$, about which one can find more information in this paper (PDF) by Baillie, Borwein, and Borwein from 2008.

Moreover, there is the binomial identity

$$\sum_{n = -\infty}^{+\infty} \binom{\alpha}{n} e^{int} = \int_{-\infty}^{+\infty} \binom{\alpha}{x} e^{itx} \, dx = (1+e^{it})^\alpha, \; \alpha >-1 \tag{2}\label{2}$$which is due to Pollard & Shisha (1973).

Finally, the authors Dominici, Gill, and Limpanuparb (2012) state the following identity involving the Bessel J function in their article (PDF)

$$\sum_{t=-\infty}^\infty \frac{J_y (at) J_y(bt)}{t}=\int_{-\infty}^\infty \frac{J_y (at) J_y(bt)}{t}\, \text{d}t. \tag{3}\label{3}$$

Product integrals

I wonder whether similar identities exist involving infinite products and their corresponding product integral. The latter is a continuous analogue of the discrete product operator.

There are multiple types of product integrals. For the purposes of this particular question, we stick to product integrals that are referred to as Type II in the wiki page referenced above:

\begin{align*} \prod_{a}^{b} f(x)^{dx} &:= \lim_{\Delta x \to 0} \prod f(x_{i})^{\Delta x} \newline &= \exp \left( \int_{a}^{b} \ln f(x) \ dx \right). \tag{4} \label{4} \end{align*}

This is called the geometric product integral.

So what I'm looking for are identities of the form

$$ \prod_{n=a}^{\infty} f(n) = \prod_{a}^{\infty} f(x)^{dx}. \tag{5}\label{5} $$

Here, $a=0$, $a=1$, or $a=-\infty$. In other words, I'm looking for identities that satisfy $$ \prod_{n=a}^{\infty} f(n) = \exp \left( \int_{a}^{\infty} \ln f(x) \ dx \right). \tag{6}\label{6}$$

If the identity holds when the limits of the product (integral) need to be shifted slightly on either side of the equation to make it work, that would also be a good example in my eyes.

Own work and Question

I've gone through a number of possibilities listed on this page. One example that comes somewhat close, but not quite, is the one associated with equation (30). It states that

$$ \prod_{n=1}^{\infty} \left(1+\frac{1}{n^3} \right) = \frac{1}{\pi} \cosh \left( \frac{1}{2} \pi \sqrt{3} \right). \tag{7}\label{7} $$ Moreover, we have

$$ \exp \left( \int_{0}^{\infty} \ln \left[ 1+ \frac{1}{x^3} \right] \ dx \right) = \exp \left( \frac{2 \pi}{ \sqrt{3}} \right). \tag{8}\label{8} $$

We know that $\cosh(x) := \frac{\exp(x)+\exp(-x)}{2}$, so it appears there are some similarities in the final expression. (Notice that, in this case, we have slightly shifted the limits of the geometric product integral.) However, \eqref{7} and \eqref{8} do no amount to the same number.

So there is more work to do. Therefore, my question is:

Are there any infinite products that are equal to their geometric product integral, thus satisfying equation \eqref{6} or some slight variant of it?

Max Muller
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  • Of the three integral equations, the first strikes me as nothing more than a mildly interesting coincidence, effectively a result substantiating the law of small numbers (expressions giving small integer values are usually equal to each other strictly by chance). But the other two apply over entire families of functions, and are thus more substantial. By using the relationship between the product and regular integrals, it should be possible to translate all three into product integral indentities. – Paul Sinclair Mar 18 '24 at 15:30
  • @PaulSinclair I've added some information on the first equation, which perhaps makes it a bit more interesting from your perspective. Above all, I am curious as to how you would employ this relation to convert the three equations into product (integral) identities – Max Muller Mar 18 '24 at 15:57
  • It is to be noted that you can "recycle" the identities with the usual integral by playing with exponentials/logarithms at the very least. One has for instance : $$ \prod_{n=-\infty}^\infty e^{\mathrm{sinc}(n)} = \exp\left(\sum_{n=-\infty}^\infty \mathrm{sinc}(n)\right) = \exp\left(\int_{-\infty}^\infty \mathrm{sinc}(x) ,\mathrm{d}x\right) = \prod_{-\infty}^\infty \left(e^{\mathrm{sinc}(x)}\right)^{\mathrm{d}x} $$ – Abezhiko Mar 18 '24 at 17:22
  • @Abezhiko I'm afraid that doesn't work, because we still require the $\ln(\cdot)$ term in the integrals – Max Muller Mar 18 '24 at 19:05
  • @MaxMuller Yes, indeed. It is present implicitly, since $\mathrm{sinc}(x) = \ln(e^{\mathrm{sinc}(x)})$. – Abezhiko Mar 19 '24 at 08:19

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