Given a series $x_i$ where $i$ is an integer, we can define $$ f(n)=\prod_{i}^n x_i $$ I am wondering if there is a way to make extend $n$ into reals? $n$ might still be finite but could be rational or irrational numbers.
Equivalently if I take a $log$ on both sides, I am also asking is there a natural way to extend the sum $$ g(n)=\sum_{i}^n y_i $$ such that $n\in \mathbb R$?