It is an obvious fact that the sum $\sum\_{n\geq 0} \binom{2n}{n} x^n$ defines an algebraic function. I am interested in the variation of this sum, namely


$$A(x)=\sum\_{n\geq 0} \binom{2n}{n}^2 x^n$$


which is not an algebraic function due to the growth of the coefficients (see Enumerative Combinatorics, Vol. 2 from Stanley, for instance).


My question is the following: can we say something about the algebraicity/transcendence of $A(x\_0)$, for any real $x\_0$ in the region of convergence)? If $x\_0$ is algebraic this is quite obvious I guess, but I do not know if there exists an argument saying something when $x\_0$ is transcendent.


I assume that this is a very general question, and an answer for any transcendent number should be very difficult. But I would be happy to know if something it can be said for some particular transcendental number ($\pi$ or $e$, for instance).


Maybe the integral representation


$$A(x)=\sum\_{n\geq 0} \binom{2n}{n}^2 x^n=\frac{1}{2\pi} \int\_{-\pi}^{\pi} \frac{du}{(1-16x\cos^2 u)^{1/2}}$$


could be helpful, and something concerning elliptic integrals can be used here, but I do not know enough on this topic.



Algebraicity of a generating function and binomial numbers
It is a straightforward computation to express $A(x) = \sum\_{n=0}^{\infty} \binom{2n}{n}^2x^n$ in terms of hypergeometric functions, namely
$$A(x) = {}\_2F\_1 (\tfrac12,\tfrac12;1;16x).$$
Then in turn one can express this function in terms of elliptic integrals,
$${}\_2F\_1 (\tfrac12,\tfrac12;1;16x) = \frac{1}{\pi} \int\_1^{\infty}\frac{du}{\sqrt{u(u-1)(u-16x)}}. \tag{1}\label{1}$$
The integral on the right is a period of the Legendre elliptic curve
$$E\_x:y^2 = u(u-1)(u-16x).$$
When $x$ is algebraic and $x \neq 0$, $1$, it is a result of Th. Schneider (1934) that such a period is transcendental over $\overline{\mathbb{Q}}$, and it was further proved by Schneider (1937) that the ratio in \eqref{1} is itself transcendental. So if $x$ is algebraic and $0<|x|< \frac{1}{16}$ (so that $A(x)$ converges), the value of $A(x)$ is transcendental over $\overline{\mathbb{Q}}$.


For an account of the connection between elliptic integrals and hypergeometric functions see Chapter 9 of Husemöller's text "[Elliptic Curves](https://doi.org/10.1007/978-1-4757-5119-2)." See Waldschmidt's survey article "[Elliptic Functions and Transcendence](https://link.springer.com/chapter/10.1007/978-0-387-78510-3_7)," Dev. Math. 17 (2008) for additional information about connections with transcendence, as well as more recent results in this area.


