dc.date.accessioned 2006-06-06T09:55:56Z dc.date.available 2006-06-06T09:55:56Z dc.date.issued 2005 dc.identifier.uri urn:nbn:de:hebis:34-2006060612903 dc.identifier.uri http://hdl.handle.net/123456789/2006060612903 dc.format.extent 115108 bytes dc.format.mimetype application/pdf dc.language.iso eng dc.subject Bieberbach-Vermutung ger dc.subject Hypergeometrische Reihe ger dc.subject Bieberbach conjecture eng dc.subject de Branges functions eng dc.subject Weinstein functions eng dc.subject hypergeometric functions eng dc.subject.ddc 510 dc.title Solution properties of the de Branges differential recurrence equation eng dc.type Preprint dcterms.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. eng dcterms.accessRights open access dcterms.creator Koepf, Wolfram dcterms.creator Schmersau, Dieter dcterms.isPartOf Mathematische Schriften Kassel ger dcterms.isPartOf 05, 18 ger dc.subject.msc 30C50 eng dc.subject.msc 33C20 eng dc.subject.swd Bieberbach-Vermutung ger
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