A binomial sum

It's not too hard to show that $$\displaystyle \binom{-\frac{1}{2}}{n} = \left( -\dfrac{1}{4} \right)^n \binom{2n}{n}$$ and hence, $$\displaystyle \frac{1}{\sqrt{1-4x}} = \sum_{n=0}^\infty \binom{-\frac{1}{2}}{n}(-4x)^n = \sum_{n=0}^\infty \binom{2n}{n}x^n$$.

Some tinkering around on WolframAlpha revealed that $$\displaystyle \sum_{n=0}^\infty \binom{3n}{n}x^n = \frac{2\cos(\frac{1}{3}\sin^{-1}(\frac{3\sqrt{3}}{2}\sqrt{x}))}{\sqrt{4-27x}}$$. By a rather natural substitution $$x = \frac{4}{27} \sin^2 \theta$$, we get

$\sum_{n=0}^\infty \binom{3n}{n}x^n = \boxed{\dfrac{\cos \frac{\theta}{3}}{\cos \theta}}.$

Sadly, WA revealed nothing about $$\displaystyle \sum_{n=0}^\infty \binom{4n}{n}x^n$$. A cool expression for $$\displaystyle \sum_{n=0}^\infty \binom{4n}{2n}x^n$$ is easily obtained with the substitution $$x = \frac{1}{16}\sin^2 \theta$$, however:

\begin{align} \sum_{n=0}^\infty \binom{4n}{2n}x^n &= \frac{1}{2} \left[ \sum_{n=0}^\infty \binom{2n}{n}(\sqrt{x})^n + \sum_{n=0}^\infty \binom{2n}{n}(-\sqrt{x})^n \right] \\ &= \frac{1}{2} \left[ \frac{1}{\sqrt{1-4\sqrt{x}}} + \frac{1}{\sqrt{1+4\sqrt{x}}} \right] \\ &= \frac{1}{2} \left[ \frac{\sqrt{1-4\sqrt{x}}+\sqrt{1+4\sqrt{x}}}{\sqrt{1-16x}} \right] \\ &= \frac{1}{2} \left[ \frac{\sqrt{1-\sin \theta}+\sqrt{1+\sin \theta}}{\cos \theta} \right] \\ &= \frac{1}{2} \left[ \frac{\sqrt{2 + 2\cos \theta}}{\cos \theta} \right] \\ &= \boxed{\dfrac{\cos \frac{\theta}{2}}{\cos \theta}}. \end{align}

Let's define the sum $$\displaystyle F_{a,b}(x) = \sum_{n=0}^\infty \binom{an}{bn}x^n$$. For convenience let's also define $$K_{a,b} = \dfrac{b^b(a-b)^{a-b}}{a^a}$$. So $$K_{2,1} = \dfrac{1}{4}$$, $$K_{3,1} = \dfrac{4}{27}$$, and $$K_{4,2} = \dfrac{1}{16}$$. It seems that the factor associated with the $$\sin^2 \theta$$ comes from the series' radius of convergence, ie

\begin{align} \binom{an}{bn} &= \frac{(an)!}{(bn)!((a-b)n)!} \\ &\sim \frac{\text{Insert Stirling's approximation}}{\text{Insert more Stirling's approximations}} \\ &\sim Cn^{-1/2} \left( \frac{a^a}{b^b(a-b)^{a-b}} \right)^n \\ &\sim Cn^{-1/2} K_{a,b}^{-n}. \end{align}

Conjecture. $$\displaystyle F_{a,b}(K_{a,b}\sin^2 \theta) = \frac{\text{Some expression}}{\cos \theta}$$.

Note by Jake Lai
5 days, 4 hours ago

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