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Hypergeometric functions in geometric function theory. (English) Zbl 0948.30024
Srinivasa Rao, K. (ed.) et al., Special functions and differential equations. Proceedings of a workshop, WSSF ’97, Madras, India, January 13-24, 1997. New Delhi: Allied Publishers Private Limited. 119-126 (1998).

A connection between special functions and a fundamental geometric property of quasiconformal maps of the unit disc, is discussed and a few of the numerous unpublished results left by S. Ramanujan in his notebooks at the time of his premature death are mentioned. Some open problems on generalized modular equations with signature $\frac{1}{a}$ and order $p$ defined by

$\frac{F\left(a,1-a;1;1-{s}^{2}\right)}{F\left(a,1-a;1;{s}^{2}\right)}=p\frac{F\left(a,1-a;1;1-{r}^{2}\right)}{F\left(a,1-a;1;{r}^{2}\right)}$

are also given.

##### MSC:
 30C62 Quasiconformal mappings in the plane 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$
##### Keywords:
asymptotic formula; quasiconformal maps