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Cancel common factors in symbolic product

Mathematica Asked by blablablup on April 1, 2021

Suppose I have the following function

G[n_, k_] := Product[g[n + i]/g[i], {i, 0, k}]

where $n,k$ are assumed to be natural numbers and $g$ is a nonzero symbolic function. I want to simplify the expression G[n,k+1]/G[n,k] and cancel all common terms to get g[n+k+1]/g[k+1]. However, Simplify, FullSimplify, Expand and Cancel all yield the symbolic (not cancelled) product.

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