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One possible answer is that hypergeometric series were (and are) used to compute periods of elliptic integrals.

In modern terminology, take a smooth cubic $X \subset \mathbb{P}^2$ whose Weierstrass form is

$y^2z=x(x-z)(x-\lambda z)y^2w=x(x-w)(x-\lambda w), \quad \lambda \in \mathbb{P}^1-\{0, 1, \infty\}$.

Then $X$ is an elliptic curve, then it can be written as $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice. It turns out that the generators $\omega_1(\lambda)$, $\omega_2(\lambda)$ of $\Lambda$, i.e. the periods of the associated Weierstrass $\wp$-function

$\wp(z; \Lambda):=\frac{1}{z^2} + \sum_{l \in \Lambda-0} \big(\frac{1}{(z-l)^2}-\frac{1}{l^2} \big)$,

can be written in terms of the standard hypergeometric series $F$, namely

$\omega_1(\lambda)=i\pi F(\frac{1}{2},\frac{1}{2} , 1, 1-\lambda)$,

$\omega_2(\lambda)=i \pi F(\frac{1}{2}, \frac{1}{2}, 1, \lambda)$.

For further details see Chapter 1 of Kobliz's book "Introduction to elliptic curves and modular forms".

2 added 282 characters in body; edited body; added 7 characters in body

One possible answer is that hypergeometric series were (and are) used to compute periods of elliptic integrals.

In modern terminology, take a smooth cubic $X$ in $\mathbb{P}^2$ X \subset \mathbb{P}^2$whose Weierstrass form is $y^2z=x(x-1)(x-\lambda), y^2z=x(x-z)(x-\lambda z), \quad \lambda \in \mathbb{P}^1-\{0, 1, \infty\}$. Then$X$is an elliptic curve, then it can be written as$\mathbb{C}/\Lambda$, where$\Lambda$is a lattice. It turns out that the generators$\omega_1(\lambda)$,$\omega_2(\lambda)$of$\Lambda$(i.e., \Lambda$, i.e. the periods of the associated Weierstrass $\wp$-function) \wp$-function$\wp(z; \Lambda):=\frac{1}{z^2} + \sum_{l \in \Lambda-0} \big(\frac{1}{(z-l)^2}-\frac{1}{l^2} \big)$, can be written in terms of the standard hypergeometric series$F$, namely$\omega_1(\lambda)=i\pi F(\frac{1}{2},\frac{1}{2} , 1, 1-\lambda)$,$\omega_2(\lambda)=i \pi F(\frac{1}{2}, \frac{1}{2}, 1, \lambda)$. For further details see Chapter 1 of Kobliz's book "Introduction to elliptic curves and modular forms". 1 One answer is periods of elliptic integrals. In modern terminology, take a smooth cubic$X$in$\mathbb{P}^2$whose Weierstrass form is $y^2z=x(x-1)(x-\lambda), \quad \lambda \in \mathbb{P}^1-\{0, 1, \infty\}$ Then$X$is an elliptic curve, then it can be written as$\mathbb{C}/\Lambda$, where$\Lambda$is a lattice. It turns out that the generators$\omega_1(\lambda)$,$\omega_2(\lambda)$of$\Lambda$(i.e., the periods of the associated Weierstrass$\wp$-function) can be written in terms of the standard hypergeometric series$F$, namely$\omega_1(\lambda)=i\pi F(\frac{1}{2},\frac{1}{2} , 1, 1-\lambda)$,$\omega_2(\lambda)=i \pi F(\frac{1}{2}, \frac{1}{2}, 1, \lambda)\$.