2

Are there a few 'nice' or 'natural' ways to motivate the existence of the Hypergeometric series

$$F(a,b;c:x) = 1 + \frac{ab}{c}x+\frac{1}{2!} \frac{a(a+1)b(b+1)}{c(c+1)}x^2+...?$$

Are there combinatorial or Physics interpretations for example?

bolbteppa
  • 4,389
  • 2
    It solves a certain differential equation; almost certainly this is how the ancients thought about it. This differential equation has some important property I don't recall; something like all differential equations of a certain type can be reduced to it by a suitable change of coordinates. This should be in the Wikipedia article. – Qiaochu Yuan Jun 29 '14 at 17:34
  • Thanks, I know the math behind it, I'm just hoping there's something beyond the standard special function usefulness idea. – bolbteppa Jun 29 '14 at 17:36
  • A lot of integrals end up giving hypergeometric functions. That's the way I think of their inception. We devised them because they show up everywhere in this context. – Cameron Williams Jun 29 '14 at 17:40
  • @bolbteppa: I don't know why 'nice' and 'natural' precludes solving a natural class of differential equations then! This is how we get many important functions, e.g. $e^x, \sin x, \cos x$... perhaps you should be more specific about your question? – Qiaochu Yuan Jun 29 '14 at 17:41
  • Probably the closest thing hypergeometric series have to a physical interpretation is in the evaluation of Feynman diagrams. – David H Jun 29 '14 at 17:46
  • Wow: "Forty-five years ago, Regge proposed that any Feynman diagram can be understood in terms of a special class of hypergeometric functions satisfying some system of differential equations so that the singularity surface of the relevant hypergeometric function coincides with the surface of the Landau singularities of the original Feynman diagram" http://quarks.inr.ac.ru/2008/proceedings/p5_FT/kalmykov.pdf or http://theor.jinr.ru/~calc2009/talk/kalmykov.pdf Keep 'em coming! – bolbteppa Jun 29 '14 at 17:51
  • @QiaochuYuan as far as I know Euler originally derived the ode from the recursion relation associated with the postulated (?) hypergeometric sequence, and because it is almost of the form $(ax^2+bx+c)y''+(cx+d)y'+ey=0$ and can be confluently-reduced to offer solutions to $(ax+b)y''+(cx+d)y'+(ex+f)y=0$ it solves many ode's whose coefficients fit into these patterns, for me it'd be more natural to take $(ax^2+bx+c)y''+(cx+d)y'+ey=0$ as the most important ode, in fact the H.G. equation looks pretty damn unnatural and special so I'm hoping the series itself has interesting interpretations. – bolbteppa Jun 29 '14 at 17:58
  • 2
    @QiaochuYuan The property you are talking about is the singularity structure: three regular singular points on the Riemann sphere. – Start wearing purple Jun 29 '14 at 18:25
  • Some more interesting avenues: "For example, in combinatorial analysis, hypergeometric identities classify single sums of products of binomial coefficients. quadratic transformations of hypergeometric functions give insight into the relationship of elliptic integrals to the arithmetic-geometric mean." http://www.smcs.num.edu.mn/cimpa/files/HG_Annarbor.pdf – bolbteppa Jun 29 '14 at 18:57

0 Answers0