Is there a way to check whether or not a certain polynomial function (of any degree) can be expressed as a binomial coefficient?
That is to say, we have the polynomial
$f(x)=n_0x^3+n_1x^2+n_3x+n_4$,
how do we know if it is possible to express $f(x)$ as$\binom{ax+b}{cx+d}$?
Additionally, if possible, how would one go about deriving the binomial expression algebraically? This is essentially working backwards from the expansion of a binomial coefficient.
Best Answer
First of all, the expression $\binom{ax+b}{cx+d}$ is never a polynomial when $c\neq 0$. This follows since $\binom{ax+b}{cx+d}\ge \big(\frac{ax+b}{cx+d}\big)^{cx+d}$, and the latter grows exponentially as $x\to\infty$ if $c\neq 0$.
Therefore, you can restrict your attention to expressions like $\binom{ax+b}{k}$. When you expand this out in powers of $x$, the highest degree term is $\frac{a^k}{k!}x^k$, and the constant term is $\binom{b}k$. Therefore, you can determine $a,b$ and $k$ by looking at the leading and constant terms of $f(x)$, plug those into $\binom{ax+b}k$, then see if the result is equal to $f(x)$. If it is, you are done, if not, then $f$ cannot be written in this form.