This thesis deals with the properties of the coefficients of Hypergeometric Series. Specifically, we are interested in which primes appear in the denominators to a bounded power. The first main result gives a method of categorizing the primes up to equivalence class which appear finitely many times in the denominators of generalized hypergeometric series nFm over the rational numbers. Necessary and sufficient conditions for when the density is zero are provided as well as a categorization of the n and m for which the problem is interesting. The second main result is a similar condition for the appearance of primes in the denominators of the hypergeometric series 2F1 over number fields, specifically quadratic extensions Q(D). A novel conjecture to the study of p-adic numbers is also provided, which discusses the digits of irrational algebraic numbers' p-adic expansions. / Thesis / Master of Science (MSc)

Thesis (Ph. D.)--Washington State University, May 2010. / Title from PDF title page (viewed on June 9, 2010). "Department of Mathematics." Includes bibliographical references (p. 232-240).

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A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / In this thesis, various partition functions with respect to Rogers-Ramanujan identities and George Andrews' partitions with initial repetitions are studied. Agarwal and Goyal gave a three-way partition theoretic interpretation of the Rogers- Ramanujan identities. We generalise their result and establish certain connections with some work of Connor. Further combinatorial consequences and related partition identities are presented. Furthermore, we re ne one of the theorems of George Andrews on partitions with initial repetitions. In the same pursuit, we construct a non-diagram version of the Keith's bijection that not only proves the theorem, but also provides a clear proof of the re nement. Various directions in the spirit of partitions with initial repetitions are discussed and results enumerated. In one case, an identity of the Euler-Pentagonal type is presented and its analytic proof given. / M T 2016

Orientador: José Plínio de Oliveira Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-15T23:12:14Z (GMT). No. of bitstreams: 1 Silva_ElenVivianiPereirada_M.pdf: 748342 bytes, checksum: 9859c0b9ff8882f29bdb000d73f74a92 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho vamos abordar dois resultados em partições planas. O primeiro, chamado Teorema Fundamental de MacMahon, nos dá uma fórmula da função geradora de partições planas de um número natural n; cuja versão da demonstração que será apresentada neste trabalho foi a prova dada por L. Carlitz em 1967. O segundo, chamado Conjectura de MacMahon, nos dá uma fórmula para a função geradora de partições planas simétricas de um número natural n, com até s níveis e com cada parte menor do que ou igual a j, este, provado por George Andrews em 1979 com um elegante argumento combinatório. Para a demonstração desses resultados usaremos identidades combinatórias e alguns resultados sobre determinantes / Abstract: In this paper we approach two results on plane partitions. The first, the MacMahon's Fundamental Theorem, gives us a formula for the generating function of plane partitions of a natural number n, whose version of the demonstration will be presented here was the proof given by L. Carlitz in 1967. The second, MacMahon's Conjecture, gives us a formula for the generating function for symmetric plane partitions of a natural number n with at most s rows and with each part at most j, this, as proven by George Andrews in 1979 with an elegant combinatorial argument. For the demonstration of these results we will use combinatorial identities and some results on determinants / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

Orientador: José Plínio de Oliveira Santos / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatisitca e Computação Cientifica / Made available in DSpace on 2018-08-16T01:34:00Z (GMT). No. of bitstreams: 1 Alegri_Mateus_D.pdf: 32503931 bytes, checksum: fb4329080c2c9c80896a52e4442b1b86 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho apresentaremos novas provas bijetivas para identidades relacionadas a partições em partes pares e distintas, generalizações das identidades de Rogers-Ramanujan entre outras. Porém o objetivo principal será trabalhar com sobrepartições de inteiros, dando a estes uma nova interpretação em termos de matrizes de três linhas. Exibiremos provas bijetivas para algumas classes de sobrepartições, apresentaremos um novo resultado que basicamente é identificar uma sobrepartição com partições planas; sendo este o principal resultado deste trabalho. No final apresentaremos algumas aplicações da representação de partição via matrizes de duas linhas: fórmulas fechadas para algumas classes destas partições. / Abstract: In this work, we present new bijective proofs for identities related to partitions into distinct even parts, generalizations of Rogers-Ramanujan identities, among others. The basic aim is to work with overpartitions of integers, give a new interpretation in terms of three-line matrices. We will show bijective proofs for some classes of overpartitions. We will present a new result that is how to identify an overpartition (with some particularities) with plane partitions; which is one of the most important results. At the end we will present some applications of the representation of a partition as a two-line array: closed formulaes for some classes of these partitions. / Doutorado / Análise Combinatória / Doutor em Matemática Aplicada

Hypergoemetric Series are very important in mathematics and come up regularly when dealing with the precise definitions of constants such as <i>e</i>, &pi; and Apery's constant &sigmaf;(3). The evaluation of such series to high precision is traditionally done with multiple divisions, multiplications and factorials, which all takes a long time to compute, especially when the computation is done on a single machine. The interest lies in performing this computation in parallel and in a distributed fashion. In this thesis, we present a simple distributed toolkit for doing such computations by splitting the problem into smaller sub-problems, solving these sub-problems in parallel on distributed machines and then combining the result at the end. Our toolkit takes care of all the networking for the user; connectivity, dropped connections, management of the Clients and the Server. All the user has to provide is the definition of the problem; how to split the problem into sub-problems, how to solve the sub-problems and finally how to combine the sub-problems and produce a result. The toolkit records timings for computation as well as for communication. What is different about our application is that all the code is written in Java (which is completely machine independent) and all the Clients are Java Applets. This means that having a web browser in enough to take part in the computation when it is distributed over the internet. We are almost guaranteed that every computer on the internet has a web browser. The Java Plug-in (if unavailable) can easily be downloaded from Sun's web site. We present a comparison between Java's native BigInteger library and an FFT based Integer Library written by R. Howell of University of Kansas. This study is important since we are doing computations with very large integers. To test our system, we evaluate <i>e</i> to different number of digits of precision and show that our system truly works and is easy for anyone to use.

Hypergoemetric Series are very important in mathematics and come up regularly when dealing with the precise definitions of constants such as <i>e</i>, &pi; and Apery's constant &sigmaf;(3). The evaluation of such series to high precision is traditionally done with multiple divisions, multiplications and factorials, which all takes a long time to compute, especially when the computation is done on a single machine. The interest lies in performing this computation in parallel and in a distributed fashion. In this thesis, we present a simple distributed toolkit for doing such computations by splitting the problem into smaller sub-problems, solving these sub-problems in parallel on distributed machines and then combining the result at the end. Our toolkit takes care of all the networking for the user; connectivity, dropped connections, management of the Clients and the Server. All the user has to provide is the definition of the problem; how to split the problem into sub-problems, how to solve the sub-problems and finally how to combine the sub-problems and produce a result. The toolkit records timings for computation as well as for communication. What is different about our application is that all the code is written in Java (which is completely machine independent) and all the Clients are Java Applets. This means that having a web browser in enough to take part in the computation when it is distributed over the internet. We are almost guaranteed that every computer on the internet has a web browser. The Java Plug-in (if unavailable) can easily be downloaded from Sun's web site. We present a comparison between Java's native BigInteger library and an FFT based Integer Library written by R. Howell of University of Kansas. This study is important since we are doing computations with very large integers. To test our system, we evaluate <i>e</i> to different number of digits of precision and show that our system truly works and is easy for anyone to use.

The present paper introduces 10 Appell’s type generalized functions Ni, i = 1, 2, ...... 10 by considering the product of n − 3F2 functions. The paper contains Fractional derivative representations, Integral representations and symbolic forms similar to those obtained by J. L. Burchnall and T. W.Chaundy for the four Appell’s functions, have been obtained for these newly defined functions N1, N2.......N10. The results obtained are believed to be new. / El presente artículo introduce 10 tipo de funciones generalizadas tipo Appell Ni, 1 ≤ i ≤ 10, considerando el producto de n funciones 3F2. El artículo contiene representaciones por derivadas fraccionales, representaciones integrales y formas simbólicas similares a aquellas obtenidas por J. L. Burchnall y T. W. Chaundy para las cuatro funciones de Appell, han sido obtenidas para estas nuevas funciones N1, N2.......N10. Los resultados parecen ser nuevos.