nLab hypergeometric function

Idea

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

Special cases appear in classical problems of mathematical physics, as solutions to the wave equation, Laplace equation or similar when 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 $\sum c_n = c_0\cdot {}_p F_q(a_1,\ldots,a_p; b_1,\ldots, b_q; x)$.

Classical case, ${}_2 F_1$

Hypergeometric function $F = {}_2 F_1(a,b;c;x)$ satisfies the ordinary differential equation

$$x(1-x)\frac{d^2 F}{d x^2} + [c-(a+b-1)x]\frac{d F}{d x} - a b F = 0.$$

For $Re(c)\gt 0$, $Re(b)\gt 0$ function ${}_2 F_1$ can be represented as the Euler integral?

$${}_2 F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^b (1-t)^{c-b-1}(1-t x)^{-a} d t,\,\,\,\,x\notin [1,+\infty).$$

The value of this function at origin is $1$. The second solution of the differential equation around $0$ is $x^{1-c} {}_2 F_1(a-c+1,b-c+1;2-c;x)$. A basis of solutions around $\infty$ is given by $x^{-a}F(a,1-c+1;1-b+a;x^{-1})$ and $x^{-b}F(b,1-c+b;1-a+b;x^{-1})$.

Confluent hypergeometric functions

If some singular points of the differential equation coalesce, in the limiting case we obtain confluent hypergeometric function (special case of which are e.g. Bessel functions).

Examples

The classical orthogonal polynomials appear as special cases for particular choices of parameters, for example Jacobi polynomials and, as their special case, Legendre polynomials.

Generalizations and related equations

There are other variants like $q$-hypergeometric functions and the basic hypergeometric series.

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, Spiridonov2013.

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 “hypergeometric” pairings between certain kind of local homology and cohomology on configuration spaces, see hypergeometric construction of KZ solutions. This has been extensively studied by Alexander Varchenko, Hiroaki Terao?, Peter Orlik, Daniel Cohen and others; often in connection to the study of (complements of) arrangements of hyperplanes in $\mathbb{C}^n$. Selberg-type integrals are involved.

References

• 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.

Lecture notes motivated by partial differential equations appearing in mathematical physics:

• Cliff P. Burgess, §10 in: Primer on Partial Differential Equations for Physicists, lecture notes (1990) [pdf, pdf]

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

• Alexander Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv. Ser. in Math. Phys. 21, World Sci. Publ. 1995. x+371 pp. (doi:10.1142/2467)

• V. Tarasov, Alexander Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pp. (arXiv:q-alg/9703044, numdam:AST_1997__246__R1_0)

Online entries/resources on hypergeometric function:

• at Wolframworld: hypergeometric function, confluent hypergeometric functon of the first kind, confluent hypergeometric functon of the second kind, generalized hypergeometric function, $q$-hypergeometric function, regularized hypergeometric function

• wikipedia: hypergeometric series, confluent hypergeometric function

• Alexander Varchenko: list of publications

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