Catalan numbers generating function

Question

I know that the generating function for the Catalan numbers sequence$$1 + 2x + 5x^2 + 14x^3 + ....$$ is $$C(x) = \frac{1 \pm \sqrt{1-4x}}{2x}$$

But I want to know why do choose $$C(x) = \frac{1 - \sqrt{1-4x}}{2x}$$

over the plus $$C(x) = \frac{1 + \sqrt{1-4x}}{2x}$$

Is there a simple explanation for that ?

Answer 1

We know from the series expansion that $C(0)=1$, so

$$1=C(0)=\lim_{x\to 0}\frac{1\pm\sqrt{1-4x}}{2x}\;.$$

If we choose the plus sign, the limit clearly doesn’t exist, since the numerator approaches $2$ while the denominator approaches $0$. Thus, we take the minus sign, so that the limit is a $\frac00$ indeterminate form, and verify that it actually is $1$.

Answer 2

The equation leading to this expression of the Catalan generating function is the following :

$$XZ^2-Z+1=0 $$

Where we need to solve this in $\mathbb{C}[[X]]$, the unkown being $Z$.

When we solve it using discriminant (the usual method for equations of degree 2), one must not forget that we de not solve it in $\mathbb{C}[[X]]$ but in $Frac(\mathbb{C}[[X]])=\mathbb{C}((X))$ the field of Laurent series.

As a reminder :

$$\text{ elements of }\mathbb{C}[[X]]\text{ are called formal power series or generating functions and a typical element is } \sum_{n\geq 0} a_nX^n $$

$$\text{ elements of }\mathbb{C}((X))\text{ are called Laurent series and a typical element is } \sum_{n\geq N} a_nX^n $$

Where $N\in\mathbb{Z}$. In other words when you have your two solutions of your equation you get :

$$Z_{\pm}(X)=\frac{1\pm \sqrt{1-4X}}{2X} $$

Now by a straightforward analysis we know that :

$$Z_-(X)\notin \mathbb{C}[[X]]\text{ and } Z_+(X)\in \mathbb{C}[[X]]$$

Since the Catalan generating function has been shown to be a solution of this equation and that it is a generating function by definition, it follows that it cannot but be $Z_+$ which is the only generating function among the solution of the equation.