# §15.17 Mathematical Applications

## §15.17(i) Differential Equations

This topic is treated in §§15.10 and 15.11.

The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. See §32.10(vi).

## §15.17(ii) Conformal Mappings

The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\Im z\geq 0$ 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 $\alpha$ and $\beta$ 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$(2,\mathbb{R})$, 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).

Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). See Andrews et al. (1999, §3.2).

## §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).