I was solving few limit questions based on factorials . They looked scary at first imagining factorials , but you take few terms out and you are done.
My question is can we find a polynomial such
$${x!}=a_nx^n+a_{n-1}x^{n-1}+…..a_0$$
We can write
$$x!=(x-(x-1)).(x-(x-2))……..(x-(x-(x-1)))$$
It can be a polynomial of maximum of x power.if it does not cancel out.
Is there a way to find coefficient of $x^k$ in this polynomial.
Best Answer
Interpreting the question as I think the author had intended, though misunderstanding the meaning of the factorial, we look at the falling Pochammersymbols $x^{\underline{n}}$ Polynomial expansion. The coefficients of this operator, known as Lah numbers. They are given by: $L(n,k)={n-1\choose k-1}\frac{n!}{k!}$. We now have the identity: $x^{\underline{n}}=\sum_{k=1}^n{(-1)^n L(n,k)x^k}$