Ordinary generating function for $\binom{3n}{n}$

Solution 1:

Standard conversion to hypergeometric series and use of the duplication and triplication formulas for the $\Gamma$-function yields \begin{equation} \sum_{n\ge 0}\binom{3n}{n}z^n = \sum_{n\ge 0}\frac{\Gamma(3n+1)}{\Gamma(2n+1)}\frac{z^n}{n!} = \sum_{n\ge 0}\frac{\Gamma(n+1/3)\Gamma(n+2/3)\Gamma(n+1)} {\Gamma(n+1/2)\Gamma(n+1)}\frac{z^n}{n!} \frac{(2\pi)^{1/2} 3^{3n+1/2}}{2\pi2^{2n+1/2}} \end{equation}

\begin{equation} = \frac{\Gamma(1/3)\Gamma(2/3)}{\Gamma(1/2)} \sum_{n\ge 0}\frac{\Gamma(n+1/3)\Gamma(n+2/3)\Gamma(1/2)} {\Gamma(n+1/2)\Gamma(1/3)\Gamma(2/3)}\frac{z^n}{n!} \frac{(2\pi)^{1/2} 3^{3n+1/2}}{2\pi2^{2n+1/2}} \end{equation}

\begin{equation} = \sqrt{3/(4\pi)} \frac{\Gamma(1/3)\Gamma(2/3)}{\Gamma(1/2)} \sum_{n\ge 0}\frac{\Gamma(n+1/3)\Gamma(n+2/3)\Gamma(1/2)} {\Gamma(n+1/2)\Gamma(1/3)\Gamma(2/3)}\frac{(2^{-2} 3^3z)^n}{n!} \end{equation}

\begin{equation} = \sqrt{3/(4\pi)} \frac{\Gamma(1/3)\Gamma(2/3)}{\Gamma(1/2)} \sum_{n\ge 0}\frac{(1/3)_n (2/3)_n} {(1/2)_n}\frac{(2^{-2} 3^3z)^n}{n!} \end{equation}

\begin{equation} = \sqrt{3/(4\pi)} \frac{\Gamma(1/3)\Gamma(2/3)}{\Gamma(1/2)}{} _2F_1(1/3, 2/3; 1/2; 2^{-2}3^3z) \end{equation} \begin{equation} = \sqrt{3} \frac{\Gamma(1/3)\Gamma(2/3)}{2\pi}{} _2F_1(1/3, 2/3; 1/2; 2^{-2}3^3z) \end{equation} Furthermore by equation 15.1.18 of Abramowitz/Stegun this Gaussian Hypergeometric Function can be reduced by \begin{equation} _2F_1(a,1-a;1/2;\sin^2z)=\frac{\cos[(2a-1)z]}{\cos z} \end{equation} with parameter $a=1/3$. Furthermore $\Gamma(1/3)\Gamma(2/3) = 2\pi/\sqrt{ 3}$ according to OEIS sequence A073006.

Solution 2:

Proposition : $$ f(z,a) = \int_{0}^{\infty} \dfrac{x^z}{x^2 +2ax +1} \mathrm{d}x = \dfrac{\pi}{\sin \pi z} \dfrac{\sin((1-z) \cos^{-1}(a))}{\sin(\cos^{-1}(a))}$$

Proof : Note that,

$$ \sum_{n=0}^{\infty}{{U}_{n}(a) {(-x)}^{n}} = \dfrac{1}{x^2 +2ax+1} $$

where $ U_{n} (x) $ is the Chebyshev Polynomial of the second kind.

$$ \implies \sum_{n=0}^{\infty}{{U}_{n}(a) \Gamma(n+1) \frac{{(-x)}^{n}}{n!}} = \frac{1}{x^2 +2ax +1} $$

Using Ramanujan Master Theorem, we have,

$$ f(z,a) = \dfrac{\pi}{\sin \pi z} U_{-s} (a) $$

$$ = \dfrac{\pi}{\sin \pi z} \dfrac{\sin((1-z) \cos^{-1}(a))}{\sin(\cos^{-1}(a))} \quad \square $$

Now, using Gamma Triplication Formula,

$$ (3n)! \; = \; \Gamma(3n+1) \; = \; \dfrac{1}{2\pi} 3^{2n+\frac12} \Gamma \left(n+\dfrac13\right)\Gamma \left(n+\dfrac23\right)\Gamma(n+1) $$

and hence

$$ {3n \choose n} \; = \; \frac{3^{3n+\frac12}}{2\pi} \times \frac{\Gamma \left(n+\dfrac13\right)\Gamma \left(n+\dfrac23 \right)}{\Gamma(2n+1)} \; = \;\frac{3^{2n+\frac12}}{2\pi} \operatorname{B} \left(n+\dfrac23,n+\dfrac13\right) $$

Thus,

$$ \begin{array}{rcl}\displaystyle \text{S} \; =\; \sum_{n=0}^\infty {3n \choose n}x^n & = & \displaystyle\frac{\sqrt{3}}{2\pi}\sum_{n=0}^\infty 3^{2n} x^n \int_0^1 u^{n-\frac13}(1-u)^{n-\frac23}\,du \\ & = & \displaystyle \frac{\sqrt{3}}{2\pi} \int_0^1 \frac{u^{-\frac13}(1-u)^{-\frac23}}{1 - 27xu(1-u)}\,du \; =\; \frac{\sqrt{3}}{2\pi}\int_0^1 \frac{u^{-\frac13}(1-u)^{-\frac23}}{1 - 4a^2\, u(1-u)}\,du \end{array} $$

where $a = \dfrac{3}{2} \sqrt{3x} $. Substituting $u = \sin^2\theta$ and then $\tan\theta = t$, we have,

$$\text{S} = \dfrac{\sqrt{3}}{2 \pi} \int_0^\infty \frac{t^{\frac{1}{3}}(1 + t^2)}{(t^2 + 2at + 1)(t^2 - 2at + 1)} \mathrm{d}t $$

Using Partial Fraction and the Proposition, we have,

$$\text{S} = \dfrac{\sqrt{3}}{2\pi} \left[ \dfrac{1}{2} f \left( \dfrac{1}{3} , -a \right) + \dfrac{1}{2} f\left( \dfrac{7}{3} , -a \right) - \dfrac{1}{4a} f\left( \dfrac{4}{3} , -a \right) - \dfrac{1}{4a} f\left( \dfrac{10}{3} , -a \right) + \dfrac{1}{2} f \left( \dfrac{1}{3} , a \right) + \dfrac{1}{2} f\left( \dfrac{7}{3} , a \right) + \dfrac{1}{4a} f\left( \dfrac{4}{3} , a \right) + \dfrac{1}{4a} f\left( \dfrac{10}{3} , a \right)\right] $$

After simplification, we have,

$$ \text{S} = \dfrac{2\cos \left(\frac{1}{3} \sin^{-1} \left(\dfrac{3\sqrt{3x}}{2} \right)\right)}{\sqrt{4-27x}} \quad \square $$