Taylor series expansion of $\frac{1}{\sqrt{1-\beta x(x+1)}}$

Question

I am trying to find the taylor series expansion about $0$ (maclaurin series) of $$x \rightarrow \frac{1}{\sqrt{1-\beta x(x+1)}} \text{ with } \beta \in \mathbb{R}^{+*}$$

I've tried using the taylor series expansion of $$\frac{1}{\sqrt{1-X}} = \sum_{n=0}^{\infty}4^{-n}{2n \choose n}X^n \text{ }\text{ }\text{ }\text{ with } \text{ } X=\beta x(x+1)$$

But I can't turn it into a power series because of the $(x+1)^n$...

I've also tried to derive $$\frac{1}{n!}\cdot\frac{\text{d}^n}{\text{d}x^n}\left(\frac{1}{\sqrt{1-\beta x(x+1)}}\right)_{x=0}$$ But no results so far...

Edit : with a more powerful method, I found that, if we call $\left(a_n\right)_{n\in\mathbb{N}}$ the coefficients of the taylor series expansion $\left(\frac{1}{\sqrt{1-\beta x(x+1)}}=\sum_{n=0}^{\infty}a_nx^n\right)$, the sequence $\left(a_n\right)_{n\in\mathbb{N}}$ is then defined by : $$\forall n\geq3, \text{} na_n=\beta\left(n-\frac{1}{2}\right)a_{n-1}+\beta\left(n-1\right)a_{n-2}$$ $$\text{with }\text{ }a_1 = \frac{\beta}{2} \text{ , } a_2 = \frac{3}{8}\beta^2+\frac{1}{2}\beta$$

It's definitely a step forward, but I don't know how to proceed from there. Is there a way to handle sequences that are defined by such a way ?

Answer 1

I would set $y = \beta x(x+1)$. With that: \begin{align} \frac1{\sqrt{1 - y}} &= \sum_{n=0}^{\infty} 4^{-n} \binom{2n}{n} y^n\\ &= \sum_{n=0}^{\infty} 4^{-n} \binom{2n}{n} \beta^n x^n (x+1)^n\\ &= \sum_{n=0}^{\infty}\sum_{k=0}^{n} 4^{-n} \binom{2n}{n} \binom{n}{k} \beta^n x^n x^k \\ &= \sum_{m=0}^{\infty} a_m x^m \end{align} where $m = n + k$, so $a_m = \sum_{n=0}^{m} 4^{-n}\binom{2n}{n}\binom{n}{m-n} \beta^n$.

$a_m$ is simply an number which you can calculate doing some sums and products. For $m = 0, \dots, 3$ the $a_m$ are $1,\frac{\beta }{2},\frac{1}{8} \beta (3 \beta +4),\frac{1}{16} \beta ^2 (5 \beta +12)$.

Mathematica "simplifies" $a_m$ to $$ a_m = 4^{-m} \binom{2 m}{m} \beta ^m \, _2F_1\left(\frac{1}{2}-\frac{m}{2},-\frac{m}{2};\frac{1}{2}-m;-\frac{4}{\beta }\right) $$ where $_2F_1$ denotes the Hypergeometric function.