From the book titled " An Introduction to Mathematical Cryptography" (Chapter 5,page 322), we know that the miller's algorithm returns a function $f_P$ whose divisor satisfies $$div(f_P) =m[P]-[mP]-(m-1)[O],$$ where $O$ is the point at infinity. When the order of $P$ is $m$, we will get $$div(f_P) =m[P]-m[O]$$.
I am thinking what if we change the input $m$ to $m^\prime=km$. Then we will get a function $f_P^\prime$ whose divisor satisfies $$div(f_P^\prime) =km[P]-[kmP]-(km-1)[O]=km[P]-km[O].$$ My question is what's the relation between the functions $f_P$ and $f_P^\prime$. Or given a point $S$, what's the relation between $f_P(S)$ and $f_P^\prime (S)$.