Generalized Binomial Biroot Conjecture

Proof for every positive integer root index, with an explicit exponential rate. Use the definition in the original source:

$$\beta_m^{(n)}(x,c) = \frac{\displaystyle \sum_{k=0}^{\lceil m/n\rceil} \binom m{nk}x^k c^{m-nk}} {\displaystyle \sum_{k=0}^{\lceil m/n\rceil-1} \binom m{nk+1}x^k c^{m-nk-1}}, \qquad x,c>0,\quad m\ge1.$$

The denominator is positive because its initial summand is $mc^{m-1}$. Terms with binomial index exceeding $m$ are zero.

First assume $n\ge2$, set $t=x^{1/n}>0$, and let $\zeta=e^{2\pi i/n}$. The numerator and denominator are, respectively,

$$N_m= \sum_{\substack{0\le j\le m\j\equiv0\;(\mathrm{mod}\;n)}} \binom mj t^jc^{m-j},$$

and

$$D_m= \frac1t \sum_{\substack{0\le j\le m\j\equiv1\;(\mathrm{mod}\;n)}} \binom mj t^jc^{m-j}.$$

Indeed,

$$\left\lceil\frac mn\right\rceil-1 = \left\lfloor\frac{m-1}{n}\right\rfloor,$$

so the denominator includes every and only nonzero residue-$1$ term. Applying the roots-of-unity filter gives the exact formulas

$$N_m=\frac1n\sum_{s=0}^{n-1}(c+t\zeta^s)^m,$$ $$D_m=\frac1{nt} \sum_{s=0}^{n-1}\zeta^{-s}(c+t\zeta^s)^m.$$

The $s=0$ summand has strictly larger modulus than all the others, because

$$|c+t\zeta^s|^2 = c^2+t^2+2ct\cos(2\pi s/n) < (c+t)^2 \qquad(1\le s<n).$$

Define

$$\rho= \frac{\sqrt{c^2+t^2+2ct\cos(2\pi/n)}}{c+t} <1.$$

Then

$$N_m=\frac{(c+t)^m}{n}(1+U_m), \qquad D_m=\frac{(c+t)^m}{nt}(1+V_m),$$

where

$$|U_m|,\ |V_m| \le(n-1)\rho^m.$$

It follows immediately that

$$\beta_m^{(n)}(x,c) = t\frac{1+U_m}{1+V_m} \longrightarrow t=x^{1/n}.$$

More precisely, whenever $(n-1)\rho^m<1$,

$$\boxed{\displaystyle \left|\beta_m^{(n)}(x,c)-x^{1/n}\right| \le \frac{2x^{1/n}(n-1)\rho^m} {1-(n-1)\rho^m}. }$$

For fixed $n,t$, the optimal spectral rate occurs precisely at $c=t$: indeed

$$\rho^2 = 1-\frac{2ct(1-\cos(2\pi/n))}{(c+t)^2},$$

and $ct/(c+t)^2\le1/4$, with equality exactly when $c=t$. The minimum contraction factor is $\cos(\pi/n)$.

The index $n=1$ must be handled separately because residues $0$ and $1$ then coincide, whereas the denominator omits its constant term. Directly,

$$N_m=(c+x)^m, \qquad D_m=\frac{(c+x)^m-c^m}{x},$$

so

$$\beta_m^{(1)}(x,c) = \frac{x}{1-(c/(c+x))^m} \longrightarrow x.$$

Thus the conjecture holds for its entire stated range $n\ge1$, $x>0$, $c>0$.

Source: Isaac Wolford, Combinatorial and Gaussian Foundations of Rational Nth Root Approximations, §3.1.3, Conjecture 3.4, https://arxiv.org/abs/2508.14095. The defining expression in that source is $\beta_m^{(n)}(x,c)$; an occurrence of $\beta_m^{(n)}(x^n,c)$ in the problem display is a transcription mismatch.