Browsing by Author "Lam, Heung Yeung"

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    The development of the elliptic functions according to Ramanujan : a thesis presented in partial fulfillment of the requirements for the degree of Master of Information Sciences in Mathematics at Massey University
    (Massey University, 2000) Lam, Heung Yeung
    Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He made substantial contributions to elliptic functions, continued fractions, infinite series, and the theory of numbers. For many years people have studied Ramanujan's work and tried to obtain a better understanding of his work. The main purpose of my thesis will be to consider some important classical results on elliptic functions and give proofs of these results using the methods which could have been used by Ramanujan. This will give an insight into how Ramanujan may have proved many of his results since his own proofs are often unknown. This thesis contains five chapters. Chapter 1 is the introduction and this is related to Chapter 2 up to Chapter 4. The goal for Chapter 2 is to write the transformation of S2n+1(q), Φr,s(q), U2n(q), and V2n(q) in terms of P(p), Q (p), and R(p). Chapter 3 discusses Ramanujan's congruence for partitions and we give a proof for Ramanujan's modulus 5 partition congruence. In Chapter 4, we investigate a method of determining the number of representations of an integer n as the sum of two, four, six, and eight squares and triangular numbers. Then we present two computer programs which are for the sums of squares and triangles. Finally, some interesting relations between the sums of squares and the sums of triangles are shown.
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    q-series in number theory and combinatorics : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand
    (Massey University, 2006) Lam, Heung Yeung
    Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He work extensively in a branch of mathematics called "q-series". Around 1913, he found an important formula which now is known as Ramanujan's 1ψ1summation formula. The aim of this thesis is to investigate Ramanujan's 1ψ1summation formula and explore its applications to number theory and combinatorics. First, we consider several classical important results on elliptic functions and then give new proofs of these results using Ramanujan's 1ψ1 summation formula. For example, we will present a number of classical and new solutions for the problem of representing an integer as sums of squares (one of the most celebrated in number theory and combinatorics) in this thesis. This will be done by using q-series and Ramanujan's 1ψ1 summation formula. This in turn will give an insight into how Ramanujan may have proven many of his results, since his own proofs are often unknown, thereby increasing and deepening our understanding of Ramanujan's work.