16 Generalized Hypergeometric Functions and Meijer $G$-FunctionApplications16.22 Asymptotic Expansions16.24 Physical Applications

- §16.23(i) Differential Equations
- §16.23(ii) Random Graphs
- §16.23(iii) Conformal Mapping
- §16.23(iv) Combinatorics and Number Theory

A variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations. These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. See, for example, Berglund et al. (1994).

A substantial transition occurs in a random graph of $n$ vertices when the number of edges becomes approximately $\frac{1}{2}n$. In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_{2}F_{2}$ functions are applied to compute the expected value of the excess.

The Bieberbach conjecture states that if ${\sum}_{n=0}^{\mathrm{\infty}}{a}_{n}{z}^{n}$ is a conformal map of the unit disk to any complex domain, then $\left|{a}_{n}\right|\le n\left|{a}_{1}\right|$. In the proof of this conjecture de Branges (1985) uses the inequality

16.23.1 | $${}_{3}F_{2}\left(\begin{array}{c}-n,n+\alpha +2,\frac{1}{2}\left(\alpha +1\right)\\ \alpha +1,\frac{1}{2}\left(\alpha +3\right)\end{array};x\right)>0,$$ | ||

when $$, $\alpha >-2$, and $n=0,1,2,\mathrm{\dots}$. The proof of this inequality is given in Askey and Gasper (1976). See also Kazarinoff (1988).

Many combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. In Petkovšek et al. (1996) tools are given for automated proofs of these identities.