# §16.23 Mathematical Applications

## §16.23(i) Differential Equations

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).

## §16.23(ii) Random Graphs

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.

## §16.23(iii) Conformal Mapping

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

 16.23.1 ${{}_{3}F_{2}}\left({-n,n+\alpha+2,\frac{1}{2}(\alpha+1)\atop\alpha+1,\frac{1}{% 2}(\alpha+3)};x\right)>0,$

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

## §16.23(iv) Combinatorics and Number Theory

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.