A generalization of Student’s t-distribution from the viewpoint of special functions

Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the Fisher F-distribution has a close relationship with the t-distribution, we also introduce a generalization of the F-distribution and prove that it converges to the chi-square distribution as the number of samples tends to infinity. Finally some particular sub-cases of these distributions are considered.

Some New Classes of Orthogonal Polynomials and Special Functions

In dieser Dissertation präsentieren wir zunächst eine Verallgemeinerung der üblichen Sturm-Liouville-Probleme mit symmetrischen Lösungen und erklären eine umfassendere Klasse. Dann führen wir einige neue Klassen orthogonaler Polynome und spezieller Funktionen ein, welche sich aus dieser symmetrischen Verallgemeinerung ableiten lassen. Als eine spezielle Konsequenz dieser Verallgemeinerung führen wir ein Polynomsystem mit vier freien Parametern ein und zeigen, dass in diesem System fast alle klassischen symmetrischen orthogonalen Polynome wie die Legendrepolynome, die Chebyshevpolynome erster und zweiter Art, die Gegenbauerpolynome, die verallgemeinerten Gegenbauerpolynome, die Hermitepolynome, die verallgemeinerten Hermitepolynome und zwei weitere neue endliche Systeme orthogonaler Polynome enthalten sind. All diese Polynome können direkt durch das neu eingeführte System ausgedrückt werden. Ferner bestimmen wir alle Standardeigenschaften des neuen Systems, insbesondere eine explizite Darstellung, eine Differentialgleichung zweiter Ordnung, eine generische Orthogonalitätsbeziehung sowie eine generische Dreitermrekursion. Außerdem benutzen wir diese Erweiterung, um die assoziierten Legendrefunktionen, welche viele Anwendungen in Physik und Ingenieurwissenschaften haben, zu verallgemeinern, und wir zeigen, dass diese Verallgemeinerung Orthogonalitätseigenschaft und -intervall erhält. In einem weiteren Kapitel der Dissertation studieren wir detailliert die Standardeigenschaften endlicher orthogonaler Polynomsysteme, welche sich aus der üblichen Sturm-Liouville-Theorie ergeben und wir zeigen, dass sie orthogonal bezüglich der Fisherschen F-Verteilung, der inversen Gammaverteilung und der verallgemeinerten t-Verteilung sind. Im nächsten Abschnitt der Dissertation betrachten wir eine vierparametrige Verallgemeinerung der Studentschen t-Verteilung. Wir zeigen, dass diese Verteilung gegen die Normalverteilung konvergiert, wenn die Anzahl der Stichprobe gegen Unendlich strebt. Eine ähnliche Verallgemeinerung der Fisherschen F-Verteilung konvergiert gegen die chi-Quadrat-Verteilung. Ferner führen wir im letzten Abschnitt der Dissertation einige neue Folgen spezieller Funktionen ein, welche Anwendungen bei der Lösung in Kugelkoordinaten der klassischen Potentialgleichung, der Wärmeleitungsgleichung und der Wellengleichung haben. Schließlich erklären wir zwei neue Klassen rationaler orthogonaler hypergeometrischer Funktionen, und wir zeigen unter Benutzung der Fouriertransformation und der Parsevalschen Gleichung, dass es sich um endliche Orthogonalsysteme mit Gewichtsfunktionen vom Gammatyp handelt.

Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

Solution properties of the de Branges differential recurrence equation

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.

Nuclear organisation and epigenetic regulation of gene expression in Dictyostelium discoideum

A series of vectors for the over-expression of tagged proteins in Dictyostelium were designed, constructed and tested. These vectors allow the addition of an N- or C-terminal tag (GFP, RFP, 3xFLAG, 3xHA, 6xMYC and TAP) with an optimized polylinker sequence and no additional amino acid residues at the N or C terminus. Different selectable markers (Blasticidin and gentamicin) are available as well as an extra chromosomal version; these allow copy number and thus expression level to be controlled, as well as allowing for more options with regard to complementation, co- and super-transformation. Finally, the vectors share standardized cloning sites, allowing a gene of interest to be easily transfered between the different versions of the vectors as experimental requirements evolve. The organisation and dynamics of the Dictyostelium nucleus during the cell cycle was investigated. The centromeric histone H3 (CenH3) variant serves to target the kinetochore to the centromeres and thus ensures correct chromosome segregation during mitosis and meiosis. A number of Dictyostelium histone H3-domain containing proteins as GFP-tagged fusions were expressed and it was found that one of them functions as CenH3 in this species. Like CenH3 from some other species, Dictyostelium CenH3 has an extended N-terminal domain with no similarity to any other known proteins. The targeting domain, comprising α-helix 2 and loop 1 of the histone fold is required for targeting CenH3 to centromeres. Compared to the targeting domain of other known and putative CenH3 species, Dictyostelium CenH3 has a shorter loop 1 region. The localisation of a variety of histone modifications and histone modifying enzymes was examined. Using fluorescence in situ hybridisation (FISH) and CenH3 chromatin-immunoprecipitation (ChIP) it was shown that the six telocentric centromeres contain all of the DIRS-1 and most of the DDT-A and skipper transposons. During interphase the centromeres remain attached to the centrosome resulting in a single CenH3 cluster which also contains the putative histone H3K9 methyltransferase SuvA, H3K9me3 and HP1 (heterochromatin protein 1). Except for the centromere cluster and a number of small foci at the nuclear periphery opposite the centromeres, the rest of the nucleus is largely devoid of transposons and heterochromatin associated histone modifications. At least some of the small foci correspond to the distal telomeres, suggesting that the chromosomes are organised in a Rabl-like manner. It was found that in contrast to metazoans, loading of CenH3 onto Dictyostelium centromeres occurs in late G2 phase. Transformation of Dictyostelium with vectors carrying the G418 resistance cassette typically results in the vector integrating into the genome in one or a few tandem arrays of approximately a hundred copies. In contrast, plasmids containing a Blasticidin resistance cassette integrate as single or a few copies. The behaviour of transgenes in the nucleus was examined by FISH, and it was found that low copy transgenes show apparently random distribution within the nucleus, while transgenes with more than approximately 10 copies cluster at or immediately adjacent to the centromeres in interphase cells regardless of the actual integration site along the chromosome. During mitosis the transgenes show centromere-like behaviour, and ChIP experiments show that transgenes contain the heterochromatin marker H3K9me2 and the centromeric histone variant H3v1. This clustering, and centromere-like behaviour was not observed on extrachromosomal transgenes, nor on a line where the transgene had integrated into the extrachromosomal rDNA palindrome. This suggests that it is the repetitive nature of the transgenes that causes the centromere-like behaviour. A Dictyostelium homolog of DET1, a protein largely restricted to multicellular eukaryotes where it has a role in developmental regulation was identified. As in other species Dictyostelium DET1 is nuclear localised. In ChIP experiments DET1 was found to bind the promoters of a number of developmentally regulated loci. In contrast to other species where it is an essential protein, loss of DET1 is not lethal in Dictyostelium, although viability is greatly reduced. Loss of DET1 results in delayed and abnormal development with enlarged aggregation territories. Mutant slugs displayed apparent cell type patterning with a bias towards pre-stalk cell types.