Contents

# Contents

## Idea

The classical hypergeometric series (introduced by Gauss) are solutions of certain ordinary differential equations of second order, the hypergeometric differential equation.

Special cases appear in classical problems of mathematical physics, solutions to the wave equation, Laplace equation or similar are attacked by Fourier method of separation of variables (cf. Legendre polynomial, Hermite polynomial).

## Definition

The hypergeometric series is defined by the formula,

${}_p F_q (a_1,\ldots,a_p; b_1,\ldots, b_q; x) = \sum_{n=0}^\infty \frac{(a_1)_n (a_2)_n\cdots (a_p)_n}{(b_1)_n (b_2)_n\cdots (b_q)_n}\frac{x^n}{n!}$

where $(a)_0 = 1$ and, for $k = 1,2,3,\ldots$

$(a)_k := a (a+1) (a+2) \cdots (a+k-1)$

is the shifted factorial. In fact let $\sum_{n = 0}^\infty c_n$ be any series of complex numbers such that $c_{n+1}/c_n$ is a rational function of $n$. Then we can find $x,p,q,a_1,\ldots,a_p,b_1,\ldots, b_q$ to write

$\frac{c_{n+1}}{c_n} = \frac{(n+a_1)(n+a_2)\cdots (n+a_p) x}{(n+b_1)(n+b_2)\cdots (n+b_q)(n+1)}$

and $\sum c_n = c_0 {}_p F_q(a_1,\ldots,a_p; b_1,\ldots, b_q; x)$.

If some singular point of the differential equation coalesce, in the limiting case we obtain confluent hypergeometric function (e.g. Bessel functions). There are other variants like $q$-hypergeometric functions and the basic hypergeometric series. The classical orthogonal polynomials appear as special cases for choices of parameters, for example Jacobi polynomials and their special case Legendre polynomials. Heun equation is the second order Fuchsian ODE with 4 regular singular points at $0,1,a$ and $\infty$; hypergeometric equation can be transformed into Heun equation by a change of variables. There is a recent elliptic version of hypergeometric functions due Spiridonov.

There are now modern generalizations to many variables due Aomoto and another variant due Mikhail Kapranov, Israel Gelfand and Andrei Zelevinsky. These multidimensional generalizations express pairings between representations of quantum groups at root of unity and representations of affine Lie algebras, which can be interpreted as pairings between certain kind of homology and cohomology on configuration spaces. This has been extensively studied by Varchenko, Terao and others; often in connection to the study of (complements of) arrangements of hyperplanes in $\mathbb{C}^n$. Selberg-type integrals are involved.

• G. E. Andrews, R. Askey, R. Roy, Special functions, Enc. of Math. and its Appl. 71, Cambridge Univ. Press 1999

• G. Gasper, M. Rahman, Basic hypergeometric series (1990)

• I. M. Gelfand, M. M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser 1994, 523 pp.

• I. M. Gel’fand, M. I. Graev, V. S. Retakh, General hypergeometric systems of equations and series of hypergeometric type, Russian Math. Surveys 47(4) (1992) 1–88 doi, transl. from Общие гипергеометрические системы уравнений и ряды гипергеометрического типа, УМН 47:4(286) (1992) 3–82 mathnet.ru; General gamma functions, exponentials, and hypergeometric functions, Russian Math. Surveys 53:1 (1998) 1–55 doi, transl. from Общие гамма-функции, экспоненты и гипергеометрические функции, УМН, 1998, 53:1 (319) 3–60 doi

• Ian G. Macdonald, Hypergeometric functions I, 1987 (arxiv/1309.4568)

• (dedicated chapter 2 of) Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, Masaaki Yoshida, From Gauss to Painlevé, A modern theory of special functions, 184 pp.

In relation to the Knizhnik-Zamolodchikov equation and quantum groups:

Online entries/resources on hypergeometric function:

There is also a far reaching elliptic generalization

• V. P. Spiridonov, Classical elliptic hypergeometric functions and their applications, pdf; Aspects of elliptic hypergeometric functions, arxiv/1307.2876