Why are hypergeometric series important and do they have a geometric or heuristic motivation? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:50:59Z http://mathoverflow.net/feeds/question/58084 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58084/why-are-hypergeometric-series-important-and-do-they-have-a-geometric-or-heuristic Why are hypergeometric series important and do they have a geometric or heuristic motivation? Qfwfq 2011-03-10T16:16:06Z 2011-03-11T19:20:19Z <p>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:</p> <blockquote> <p>What are hypergeometric series really, and why they should be (or have been in the past centuries) important/interesting?</p> </blockquote> http://mathoverflow.net/questions/58084/why-are-hypergeometric-series-important-and-do-they-have-a-geometric-or-heuristic/58085#58085 Answer by Francesco Polizzi for Why are hypergeometric series important and do they have a geometric or heuristic motivation? Francesco Polizzi 2011-03-10T16:28:17Z 2011-03-10T17:00:55Z <p>One possible answer is that hypergeometric series were (and are) used to compute <em>periods of elliptic integrals</em>.</p> <p>In modern terminology, take a smooth cubic $X \subset \mathbb{P}^2$ whose Weierstrass form is</p> <p><code>$y^2w=x(x-w)(x-\lambda w), \quad \lambda \in \mathbb{P}^1-\{0, 1, \infty\}$</code>.</p> <p>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 </p> <p>$\wp(z; \Lambda):=\frac{1}{z^2} + \sum_{l \in \Lambda-0} \big(\frac{1}{(z-l)^2}-\frac{1}{l^2} \big)$,</p> <p>can be written in terms of the standard hypergeometric series $F$, namely</p> <p>$\omega_1(\lambda)=i\pi F(\frac{1}{2},\frac{1}{2} , 1, 1-\lambda)$, </p> <p>$\omega_2(\lambda)=i \pi F(\frac{1}{2}, \frac{1}{2}, 1, \lambda)$.</p> <p>For further details see Chapter 1 of Kobliz's book "Introduction to elliptic curves and modular forms".</p> http://mathoverflow.net/questions/58084/why-are-hypergeometric-series-important-and-do-they-have-a-geometric-or-heuristic/58089#58089 Answer by J.C. Ottem for Why are hypergeometric series important and do they have a geometric or heuristic motivation? J.C. Ottem 2011-03-10T16:46:26Z 2011-03-11T19:20:19Z <p>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 $$z=t {}_4 F_3(\frac15,\frac25,\frac35,\frac45,\frac12,\frac34,\frac54,\frac{5^5}{4^4}t^4)=t+t^5+10\frac{ t^9}{2!}+15\cdot 14 \frac{t^{13}}{3!}+\ldots$$</p> http://mathoverflow.net/questions/58084/why-are-hypergeometric-series-important-and-do-they-have-a-geometric-or-heuristic/58090#58090 Answer by JME for Why are hypergeometric series important and do they have a geometric or heuristic motivation? JME 2011-03-10T16:47:02Z 2011-03-10T17:08:39Z <p>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. </p> <p>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. </p> <p>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. </p> <p>I guess you can read more about them in <A href="http://en.wikipedia.org/wiki/Hypergeometric_function" rel="nofollow"> this wikipedia page</A> or in these <A href="http://math.georgiasouthern.edu/~asills/teach/spr05/hypergeom.pdf" rel="nofollow"> notes</A>. Several examples of applications to number theory, physics and combinatorics can be read <A href="http://www.math.psu.edu/andrews/pdf/59.pdf" rel="nofollow"> here </A>.</p> http://mathoverflow.net/questions/58084/why-are-hypergeometric-series-important-and-do-they-have-a-geometric-or-heuristic/58183#58183 Answer by none for Why are hypergeometric series important and do they have a geometric or heuristic motivation? none 2011-03-11T18:18:15Z 2011-03-11T18:18:15Z <p>You might like the wonderful book A=B:</p> <p><a href="http://www.math.upenn.edu/~wilf/AeqB.html" rel="nofollow">http://www.math.upenn.edu/~wilf/AeqB.html</a></p> http://mathoverflow.net/questions/58084/why-are-hypergeometric-series-important-and-do-they-have-a-geometric-or-heuristic/58187#58187 Answer by Jacques Carette for Why are hypergeometric series important and do they have a geometric or heuristic motivation? Jacques Carette 2011-03-11T19:05:02Z 2011-03-11T19:05:02Z <p>The (general) hypergeometric equation has one more property which has not yet been mentionned: it has as (formal) solution at 0 <strong>exactly</strong> a series whose sequence of coefficients satisfies a <em>first-order</em> linear recurrence equation with polynomial coefficients. [One has to include all pFq here, including divergent ones].</p> <p>The result above is a formal version of the action of the inverse Mellin transform on linear recurrence equations with polynomial coefficients.</p> <p>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.</p> <p>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.</p>