Solution properties of the de Branges differential recurrence equation

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 published by Askey and Gasper in 1976 (Askey, R. and Gasper, G., 1976, Positive Jacobi polynomial sums II. American Journal of Mathematics, 98, 709–737.). The de Branges functions are defined as the solutions of a system of differential recurrence equations with suitably given initial values. The essential fact used in the proof of the Bieberbach and Milin conjectures is the statement . 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 directly connected with de Branges’, , and the positivity results in both proofs are essentially the same. In this article we study differential recurrence equations equivalent to de Branges’ original ones and show that many solutions of these differential recurrence equations don’t change sign so that the above inequality is not as surprising as expected. Furthermore, we present a multiparameterized hypergeometric family of solutions of the de Branges differential recurrence equations showing that solutions are not rare at all.