Preprint
Solution properties of the de Branges differential recurrence equation
Abstract
In this 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 thas was published by Askey and Gasper in 1976. The de Branges functions Tn/k(t) 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 Tn/k(t)<=0. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λn/k(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', Tn/k(t)=-kΛn/k(t), and the positivity results in both proofs Tn/k(t)<=0 are essentially the same. In this paper 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.
Citation
@article{urn:nbn:de:hebis:34-2006060612903,
author={Koepf, Wolfram and Schmersau, Dieter},
title={Solution properties of the de Branges differential recurrence equation},
year={2005}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2005$n2005 1500 1/eng 2050 ##0##urn:nbn:de:hebis:34-2006060612903 3000 Koepf, Wolfram 3010 Schmersau, Dieter 4000 Solution properties of the de Branges differential recurrence equation / Koepf, Wolfram 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-2006060612903=x R 4204 \$dPreprint 4170 Mathematische Schriften Kassel ;; 05, 18 5550 {{Bieberbach-Vermutung}} 7136 ##0##urn:nbn:de:hebis:34-2006060612903
2006-06-06T09:55:56Z 2006-06-06T09:55:56Z 2005 urn:nbn:de:hebis:34-2006060612903 http://hdl.handle.net/123456789/2006060612903 115108 bytes application/pdf eng Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Bieberbach-Vermutung Hypergeometrische Reihe Bieberbach conjecture de Branges functions Weinstein functions hypergeometric functions 510 Solution properties of the de Branges differential recurrence equation Preprint In this 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 thas was published by Askey and Gasper in 1976. The de Branges functions Tn/k(t) 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 Tn/k(t)<=0. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system Λn/k(t) which (by Todorov and Wilf) was realized to be directly connected with de Branges', Tn/k(t)=-kΛn/k(t), and the positivity results in both proofs Tn/k(t)<=0 are essentially the same. In this paper 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. open access Koepf, Wolfram Schmersau, Dieter Mathematische Schriften Kassel ;; 05, 18 30C50 33C20 Bieberbach-Vermutung Mathematische Schriften Kassel 05, 18
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