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I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied.

Here is my question:

For every $K$ being an odd number, the polynomial:

$$\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left(2(1-\theta)^{2K-i}-(1-2\theta)^{K-i}\right)=0$$

has a single root $\theta_0$ in $\theta\in(0,\,0.5)$ and when $K$ increases $\theta_0$ monotonically decreases to zero, i.e.,

$$\lim_{K\rightarrow \infty}\theta_0=0$$

Is there any way to deal with this problem to establish a proof that indeed the claim is true?


The polynomial above is the manipulation of the difference of two binomial c.d.f.s in the following way:

$$B(K/2;K,1-x)|_{x=\theta}-(1/2)B(K/2;K,1-x)|_{x=\theta/(1-\theta)}$$

I use the identity $$B(K/2;K,1-x)=1-B(K/2;K,x)=\sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}x^i(1-x)^{K-i}$$

Added (01.10): I simplified it using regularized incomplete beta function. The roots of the given polynomial is equal to the roots of

$$\small 2\theta^{\frac{1+K}{2}} F_1\left(\frac{1-K}{2},\frac{1+K}{2},\frac{3+K}{2},\theta\right)-\left(\frac{\theta}{1-\theta}\right)^{\frac{1+K}{2}} F_1\left(\frac{1-K}{2},\frac{1+K}{2},\frac{3+K}{2},\frac{\theta}{1-\theta}\right)=0$$

where $F_1$ is the hypergeometric function with respective parameters (how mathematica uses)

Thank you very much in advance.

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  • $\begingroup$ A standard technique for counting real roots in an interval is en.wikipedia.org/wiki/Sturm%27s_theorem $\endgroup$ Oct 1, 2014 at 6:52
  • $\begingroup$ I doubt that Sturm's Theorem will be helpful here. $\endgroup$ Oct 1, 2014 at 9:01
  • $\begingroup$ @PeterMueller I read Sturm's theorem. I think it is interesting but hardly helpful here. The difficulty is apparently on the fact that for every $K$ the polynomial that we have is different and the total number of roots of the polynomial is also increasing with $K$, Sturm's theorem hardly useful. I think there should be some generic pattern in the definition of the polynomials, therefore I added some more information. $\endgroup$ Oct 1, 2014 at 10:16
  • $\begingroup$ Maybe some effective version of the Moivre-Laplace Theorem (approximating a binomial distribution by a normal distribution) could help here. $\endgroup$ Oct 1, 2014 at 11:34
  • $\begingroup$ @PeterMueller I was thinking in the same line. Those polynomials are related to large deviations theory for Binomial r.v.s and there is an exponential decay rate. But one has to guarantee that the effectiveness of approximation is good enough s.t. monotonicity of $\theta_0$ is not affected with the modeling errors. $\endgroup$ Oct 1, 2014 at 11:55

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