Background
The Euler-Maclaurin (E-M) formula is a formula for the difference between the sum and the integral of a real or complex continuous function on the interval $[m,n]$. It expresses this difference in terms of the higher derivaties $f^{(k)} (x)$.
More specifically, the E-M formula states that, for a $p$-times differentiable function $f(\cdot)$, we have $$ \sum_{i=m}^{n} f(i) - \int_{m}^{n} f(x) \ dx = \frac{f(n)+f(m)}{2} + \sum_{k=1}^{ \lfloor \frac{p}{2} \rfloor } \frac{B_{2k}}{(2k)!} \left( f^{(2k-1)} (n) - f^{(2k-1)} (m) \right) + R_{p}. \label{1}\tag{1} $$
Here, $B_{k}$ is the $k$'th Bernoulli number, and $R_{p}$ is an error term that depends on $n,m,p,$ and $f$.
There are many generalizations of the E-M formula. Some of these are described in the following paper by Sarafyan et al. (1979). Also, this article by Karshon et al. (2007) discusses Euler-Maclaurin formulas for lattice polytopes. However, I haven't found or obtained any analogous formulas for products yet.
Product Integrals
I wonder whether a similar formula exists that expresses the difference between a product and a product integral of a function as some expression involving the higher derivates of that function. The product integral is the continuous analogue of the discrete product operator.
There are three types of product integrals. For the purposes of this question, it is perhaps convenient to restrict our focus to product integrals of Type II. These are called geometric integrals. They are defined as follows:
$$ \prod_{a}^{b} f(x)^{dx} := \exp \left( \int_{a}^{b} \ln f(x) \ dx \right) . \tag{2}\label{2} $$
I am not sure whether we should be looking for a formula describing the difference between a product and its corresponding product integral, or one that describes the ratio of a product and the product integral. The first variant would look most like the classical E-M formula, but the second variant may be a more natural expression to consider, since division is the inverse of the product.
Question
I've tried adjusting formula \eqref{1} to find some expression for a product analogue of the E-M formula, by setting $g(i) := \exp(f(i))$ and taking logs of both sides of the equation. However, even though one obtains a product $ \prod_{i=m}^{n} g(i) $ this way, one does not obtain a geometric product integral when one applies this procedure to classical integral. I am therefore not sure how to obtain such a formula myself, and I also haven't found it in the literature. So my question is:
Is there an E-M-like formula that relates products and (geometric) product integrals with one another?
(Partial answer, too long for a comment, etc.)
The proof behind Euler-Maclaurin formula is based on the properties of Bernoulli polynomials (with respect to differentiation among others), combined with iterated integrations by parts. If you plan to construct a multiplicative analog, several remarks have to be treated.
Firstly, the major difficulty in order to do so lies in the integration by parts in fact. Indeed, when applied repeatedly, it permits to add new terms to the expansion in order to make it more accurate; actually, these new terms are the boundary terms, while the remainder contains the higher-order derivatives of the considered function.
However, the geometric analog of integration by parts doesn't exist for product integral. Indeed, one has $\prod (f(x)g(x))^{\mathrm{d}x} = \prod f(x)^{\mathrm{d}x} \prod g(y)^{\mathrm{d}y}$, which corresponds to the property of linearity for the standard (additive) integral. This relation is itself due to the absence of a Leibniz rule for the geometric derivative, since $(fg)_* = f_*g_*$, where $f_* := \exp\left(\frac{\mathrm{d}}{\mathrm{d}x} \ln f\right)$ denotes the said geometric derivative. In conclusion, you would have to find another technique of integration as a replacement, which would produce a product expansion by multiplying successive boundary terms.
Secondly, within the classical Euler-Maclaurin formula, the boundary terms come with Bernoulli numbers, or more precisely, with Bernoulli polynomials evaluated on the boundary. In the product case, the boundary terms generated by your new technique of integration (replacing integration by parts) will be probably accompanied by a sequence of (not necessarily polynomial) functions with specific properties, in particular with respect to the geometric derivative, which will play the analogous role of Bernoulli polynomials.
In order to describe this family of functions correctly, you will need to define the multiplicatively equivalent notions of generating functions, Taylors series and so forth. In conclusion, I suspect it is possible to construct an analog of the Euler-Maclaurin formula for products, but a lot of work has to be done beforehand, because one starts almost from scratch.