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

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 ?

## 1 Answer

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.

• Well actually, after numerical simulation, your numerical values don't seem to work... – Harmonic Sun Dec 14 '18 at 0:17
• @HarmonicSun Yes, I had forgotten to include $\beta$. The values are not as pretty now but you can still easily calculate them. – Carlos Esparza Dec 14 '18 at 0:20
• Ok thanks a lot ! Additionaly, do you think that based on my edit on my initial question, one can derive a prettier closed form for the $a_m$ ? – Harmonic Sun Dec 14 '18 at 0:26
• I might be picky but, in your formula for $a_m$, ${n \choose m-n}$ is not defined for, say, n=0... Or do we take it a 0 conventionaly ? – Harmonic Sun Dec 14 '18 at 0:33
• @HarmonicSun Yes, the Binomial coefficient $\binom{a}{b}$ is understood to be zero when $a < b$ (and $a, b \in \mathbb{Z}$). – Carlos Esparza Dec 14 '18 at 0:39