§15.17 Mathematical Applications
Contents
- §15.17(i) Differential Equations
- §15.17(ii) Conformal Mappings
- §15.17(iii) Group Representations
- §15.17(iv) Combinatorics
- §15.17(v) Monodromy Groups
§15.17(i) Differential Equations
§15.17(ii) Conformal Mappings
The quotient of two solutions of (15.10.1) maps the closed upper
half-plane conformally onto a curvilinear triangle. See
Klein (1894) and Hochstadt (1971). Hypergeometric
functions, especially complete elliptic integrals, also play an important role
in quasiconformal mapping. See Anderson et al. (1997).
§15.17(iii) Group Representations
For harmonic analysis it is more natural to represent hypergeometric functions
as a Jacobi function (§15.9(ii)). For special values of and
there are many group-theoretic interpretations. First, as spherical
functions on noncompact Riemannian symmetric spaces of rank one, but also as
associated spherical functions, intertwining functions, matrix elements of
SL
, and spherical functions on certain nonsymmetric Gelfand pairs.
Harmonic analysis can be developed for the Jacobi transform either as a
generalization of the Fourier-cosine transform (§1.14(ii)) or as a
specialization of a group Fourier transform. For further information see
Koornwinder (1984a).
§15.17(iv) Combinatorics
In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. See Egorychev (1984, §2.3).
§15.17(v) Monodromy Groups
The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. For a survey of this topic see Gray (2000).