An investigation into modular forms, Q-series, partitions and related applications

Abstract

This thesis presents some new identities between Ramanujan's arithmetical function r(n) and the divisor functions σk(n) = Σdln d^k. Known congruence properties of r(n) are used to derive an upper bound M for which it can be shown that r(n) ≠ 0 for all n ≤ M.

Jacobi's Triple Product Identity and the Quintuple Product Identity along with the Chebyshev Polynomials are used to derive many summation theorems in real α and β, where αbeta = ±1. It is shown how these can be applied to sums of reciprocals of Fibonacci and Lucas numbers, to produce many new and interesting identities. In fact these results are applicable to any sequence of numbers defined by a second order linear recurrence relation of the form U_n_+1 = λU_n-1. It is shown how the well known modular transformations of the standard theta functions θ2, θ3 and θ4 can be used to produce similar results.

Many new and beautiful results of the previous character are arrived at using a remarkable elementary idea, without the help of the more advanced theory of elliptic functions used in the derivation of earlier results. Agains, these theorems are applicable to any sequence defined by the above second order linear recurrence relation.

Some new polynomial identities involving Fibonacci and Lucas numbers are derived using Chebyshev-like polynomials. These include a generalisation of the well known identity F_{3n =} F_n(5F_n^{2 +} 3(-1)ⁿ).

Finally some previous results are applied to the theory of highly restricted partitions, identities involving sums of binomial coefficients and representations of the unrestricted partition function.

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