Weinstein's functions and the Askey-Gasper identity

Abstract

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realized to be the same as de Branges'. In this article, we show how a variant of the Askey-Gasper identity can be deduced by a straightforward examination of Weinstein's functions which intimately are related with a L\"owner chain of the Koebe function, and therefore with univalent functions.