nLab
hypergeometric function

Classical hypergeometric series (introduced by Gauss) are solutions of certain ordinary differential equation of the second order; special cases appear in classical problems of mathematical physics, when wave, Laplace and similar equations are attacked by Fourier method of separation of variables (cf. Legendre polynomial, Hermite polynomial?). Hypergeometric series is defined by the formula,

$${}_p{F}_q(a_1,\dots ,a_p;b_1,\dots ,b_q;x)=\sum _{n=0}^{\mathrm{\infty }}\frac{(a_1{)}_n(a_2{)}_n\cdots (a_p{)}_n}{(b_1{)}_n(b_2{)}_n\cdots (b_q{)}_n}\frac{x^n}{n!}$${}_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 $k=1,2,3,\dots$

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

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

$$\frac{c_{n+1}}{c_n}=\frac{(n+a_1)(n+a_2)\cdots (n+a_p)x}{(n+b_1)(n+b_2)\cdots (n+b_q)(n+1)}$$\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 $\sum c_n=c_0{}_p{F}_q(a_1,\dots ,a_p;b_1,\dots ,b_q;x)$.

There are variants like the confluent hypergeometric function (e.g. wikipedia:Bessel function), $q$-hypergeometric functions and the basic hypergeometric series. The classical orthogonal polynomials appear as special cases for choices of parameters. There is a recent elliptic version 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 homlogy 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 ${ℂ}^n$. 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.

• A. 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.

• V. Tarasov, A. Varchenko, Geometry of $q$-hypergeometric functions, quantum affine algebras and elliptic quantum groups, Astérisque 246 (1997), vi+135 pp.

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

• V. P. Spiridonov, Classical elliptic hypergeometric functions and their applications, pdf