How to transform a definite a nested product into nested indefinite products?

Question

A nested product is said to be indefinite if its multiplicand is free of all of its upper bound variables. Otherwise, it is said to be a definite product. For example, $$\prod_{i=1}^{n}\prod_{j=1}^{i}\prod_{k=1}^{j}\frac{k^{2}}{2\,k+1}$$ is a nested indefinite product since its multiplicand, $\frac{k^{2}}{2\,k+1}$ is free of all the upper bound variables $n, i, j$. However, $$\prod_{i=1}^{n}\prod_{j=1}^{i}\prod_{k=1}^{j}\frac{i+j+k}{i+j+k+2}$$ is a definite product since its multiplicand $\frac{i+j+k}{i+j+k+2}$ has the upper bound variables $i$ and $j$. Is there a general method of transforming a (nested) definite product to an indefinite (nested) product or a product of indefinite (nested) products?