Question 168
Define the polynomials $\binom{x}{j} = \frac{x(x-1)(x-2)\cdots(x-j+1)}{j!}$ for $j$ positive integer and $\binom{x}{0} = 1$.
a. Prove that all non-nil polynomials can be written uniquely as a linear combination of such polynomials $\binom{x}{n}$.
b. Let $c(n,k)$ be the coefficient of $\binom{x}{k}$ in $x^n$ (as described in the previous item). Compute
$$
\frac{c(n, k) + c(n, k + 1)}{c(n + 1, k + 1)}.
$$
a. Prove that all non-nil polynomials can be written uniquely as a linear combination of such polynomials $\binom{x}{n}$.
b. Let $c(n,k)$ be the coefficient of $\binom{x}{k}$ in $x^n$ (as described in the previous item). Compute
$$
\frac{c(n, k) + c(n, k + 1)}{c(n + 1, k + 1)}.
$$