One Russian book

[rsoldo]

For $m,n\in\mathbb{N}$ determine polynomials $M$ and $N$ that:
$x^{m}M(x)+(1-x)^{n}N(x)=1$.

Result: Divide with $(1-x)^{n}$ and derive $m-1$ times.
$N(x)=1+\frac{n}{1}x+\frac{n(n+1)}{1\cdot2}x^{2}+\ldots+\frac{n(n+1)\ldots (n+m-2)}{1\cdot2\ldots(m-1)}x^{m-1};$
$M(x)=\frac{(m+1)(m+2)\ldots(m+n-1)}{(n-1)!}-\frac{m}{1}\cdot\frac{(m+2)\ldots(m+n-1)}{(n-2)!}x+\frac{m(m+1)}{1\cdot2}\cdot\frac{(m+3)\ldots(m+n-1)}{(n-3)!}x^{2}-\ldots+(-1)^{n-1}\frac{m(m+1)\ldots(m+n-2)}{(n-1)!}x^{n-1}.$