Bernouli polynomials satisfies the relation $B_n(x+1)-B_n(x)=nx^{n-1}$.
Can we write $B_n(x)-B_n(0)=nx^{n-1}$ or something like $B_{n+1}(x)-B_n(0)$ to be equal to $nx^{n-1}$?
I mean I want to change the subscripts to be $B_{n+1}$ and $B_n$ instead of keeping both subscripts to be $B_n$.
Well, $B_2(x) = x^2 - x - 1/6$, so $B_2(x) - B_2(0) = x^2 - x \neq 2x$.
Maybe you're looking for something like this: $$\sum_{i=0}^x i^n = \frac{B_{n+1}(x+1) - B_{n+1}(0)}{n+1}.$$