Lanterna Rally

Author: Eduardo Evans

Publisher:

Published: 2023

Total Pages: 0

ISBN-13:

Preliminary identities in the theory of basic hypergeometric series, or `q-series', are proven. These include q-analogues of the exponential function, which lead to a fairly simple proof of Jacobi's celebrated triple product identity due to Andrews. The Dedekind eta function is introduced and a few identities of it derived. Euler's pentagonal number theorem is shown as a special case of Ramanujan's theta function and Watson's quintuple product identity is proved in a manner given by Carlitz and Subbarao. The Jacobian theta functions are introduced as special kinds of basic hypergeometric series and various relations between them derived using the triple product identity, among other previously established results. A special quotient of theta functions is introduced as the modular lambda function. The Eisenstein series are first defined through their Lambert series expansions and a series of differential equations due to Ramanujan are developed. Modular forms and functions and subsequently elliptic functions are introduced. The Weierstrass p-function is developed along other elliptic functions, those being defined as certain quotients of theta functions. The first few Eisenstein series are then shown to be expressible in terms of theta functions. Theta functions are shown to be related to Gauss' hypergeometric series _2F_1(a,b;c;z) through the Jacobi inversion theorem. This is shown to have use in relating modular equations and hypergeometric series to pi. The arithmetic-geometric mean iteration of Gauss is developed and used in conjunction with other results established in proofs of two iterative algorithms for pi. Recent applications of pi algorithms using and not using the techniques developed here are then discussed.