# Why are hypergeometric series important and do they have a geometric or heuristic motivation?

Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia article doesn't illuminate much about why this kind of special functions should form such a natural topic in mathematics (and in fact have been throughout 19th century). Simply:

What are hypergeometric series really, and why they should be (or have been in the past centuries) important/interesting?

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Hypergeometric series are solutions of a large class of differential equations. A series $\sum_{k} a_k t^k$ is hypergeometric if $Q_{k}=\frac{a_{k+1}}{a_k}$ is a rational function. Many familiar functions (trigonometric functions, exponential,logarithm,Hermite polynomials, Laguerre polynomials, etc) are hypergeometric.

Hypergeometric functions show up as solutions of many important ordinary differential equations. In particular in physics, for example in the study of the hydrogene atom (Hermite polynomials) and in simple problems of classical mechanics.

Hypergeometric functions are also important in the study of elliptic elliptic curves where they can be used to compute the inverse of the $j$-invariant.

I guess you can read more about them in this wikipedia page or in these notes. Several examples of applications to number theory, physics and combinatorics can be read here .

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In the 19th century, a lot of efforts were made in order to solve the general quintic equation $x^5+a_4x^4 +a_3x^3 +a_2x^2 +a_1x +a_0$ using special functions. It turns out that the roots of this equation are expressible in terms of hypergeometric series. To wit, one possibility is by first reducing the number of parameters, to the form $x^5-x-t=0$. Then a Lagrange inversion argument essentially gives a root $$z=t {}_4 F_3(\frac15,\frac25,\frac35,\frac45,\frac12,\frac34,\frac54,\frac{5^5}{4^4}t^4)=t+t^5+10\frac{ t^9}{2!}+15\cdot 14 \frac{t^{13}}{3!}+\ldots$$

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For a combinatorial interpretation of this series solution see qchu.wordpress.com/2010/10/08/… . –  Qiaochu Yuan Mar 10 '11 at 16:58
I should point out that the connection between hypergeometric functions and algebraic equations is more than a pure coincidence of series inversion. Felix Klein's 'Lectures on the Icosahedron' offers a very nice derivation of the roots from a geometric viewpoint, using the symmetry group of the icosahedron and a Galois resolvent of degree 60. –  J.C. Ottem Mar 10 '11 at 17:11

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

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

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".

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The (general) hypergeometric equation has one more property which has not yet been mentionned: it has as (formal) solution at 0 exactly a series whose sequence of coefficients satisfies a first-order linear recurrence equation with polynomial coefficients. [One has to include all pFq here, including divergent ones].

The result above is a formal version of the action of the inverse Mellin transform on linear recurrence equations with polynomial coefficients.

In other words, structurally speaking, the hypergeometric equation is 'first order' (because its coefficient sequence is, not because its differential equation is), but it is the most general such.

The fact that this corresponds to a lot of known functions, as well as showing up in quite a few other places, is (often) a reflection of these structural properties.

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