Solving equation involving binomial function

Question

Solve for $x$ in terms of $i$ and $j$:

$$ \binom{x}{i} = j $$

where $x$ is Real; $i$ and $j$ are Integers: $x \geqslant i$, $i \geqslant1$, $j \geqslant 0$.

I came across this problem while trying to unrank combinations $\binom{n}{k}$, pp. 51, Algorithm 2.12. I tried the following:

$$ Minimize \left | \binom{x}{i}-j \right | $$

$$ s.t. x\geqslant i $$

A plot of $|\binom{x}{i}-j|$ for $i=20$ and $j=10$ is the following:

I ignore if there is a closed-form for the real solutions of $x$ in terms of $i$ and $j$.

Answer 1

$\displaystyle{x\choose i}$ is a polynomial of degree i in x. For $i<5$, we have the linear, quadratic, cubic, and quartic formulas for roots. Otherwise, the Abel-Ruffini theorem informs us that no such general formula exists. $~$ Nevertheless, the rational root theorem or other polynomial factorization methods might prove helpful. However, I am afraid that recourse to numerical methods is ultimately unavoidable.

Answer 2

As a partial answer I generated some solutions using CAS.

For $i=1$: $x_1 = j.$

For $i=2$:

$$x_1 = 1/2+1/2\,\sqrt {1+8\,j},$$ $$x_2 = 1/2-1/2\,\sqrt {1+8\,j}.$$

For $i=3$:

$$x_1 = 1/3\,\sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}+{\frac {1}{\sqrt [3]{ 81\,j+3\,\sqrt {729\,{j}^{2}-3}}}}+1,$$

$$x_2 = -1/6\,\sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}-1/2\,{\frac {1}{ \sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}}}+1+1/2\,i\sqrt {3} \left( 1/3\,\sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}-{\frac {1}{\sqrt [3]{ 81\,j+3\,\sqrt {729\,{j}^{2}-3}}}} \right),$$

$$x_3 = -1/6\,\sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}-1/2\,{\frac {1}{ \sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}}}+1-1/2\,i\sqrt {3} \left( 1/3\,\sqrt [3]{81\,j+3\,\sqrt {729\,{j}^{2}-3}}-{\frac {1}{\sqrt [3]{ 81\,j+3\,\sqrt {729\,{j}^{2}-3}}}} \right).$$

For $i=4$:

$$x_1 = 3/2+1/2\,\sqrt {5+4\,\sqrt {1+24\,j}},$$

$$x_2 = 3/2-1/2\,\sqrt {5+4\,\sqrt {1+24\,j}},$$

$$x_3 = 3/2+1/2\,\sqrt {5-4\,\sqrt {1+24\,j}},$$

$$x_4 = 3/2-1/2\,\sqrt {5-4\,\sqrt {1+24\,j}}.$$

For $i=5$ I didn't find closed-form.

And the solutions for $i=6$ are here.

I'm also very interested in a general solution.