Modular forms and two new integer sequences at level 7 : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand

Abstract

Integer sequences resulting from recurrence relations with polynomial coefficients are rare. Two new integer sequences have been discovered and are the main result in this thesis. They consist of a three-term quadratic recurrence

(𝑛+1)²𝑐₇(𝑛+1) = (26𝑛² + 13𝑛 + 2)𝑐₇(𝑛) + 3(2𝑛 - 2)𝑐₇(𝑛-1)

with initial conditions 𝑐₇(-1) = 0 and 𝑐₇(0) = 1, and a five-term quartic recurrence

(𝑛 + 1)⁴𝘶₇(𝑛 + 1) = -𝑃𝘶₇(𝑛) - 𝘘𝘶₇(𝑛 - 1) - 𝘙𝘶₇(𝑛-2) - 𝘚𝘶₇(𝑛 - 3)

where

𝑃 = 26𝑛⁴ + 52𝑛³+ 58𝑛² + 32𝑛 + 7, 𝘘 = 267𝑛⁴ + 268𝑛² + 18, 𝘙 = 1274𝑛⁴ - 2548𝑛³ + 2842𝑛² - 1568𝑛 + 343, 𝘚 = 2401(𝑛 - 1)⁴

with initial conditions 𝘶₇(0) = 1 and 𝘶₇(-1) = 𝘶₇(-2) = 𝘶₇(-3) = 0. The experimental procedure used in their discovery utilizes the mathematical software Maple. Proofs are given that rely on the theory of modular forms for level 7, Ramanujan's Eisenstein series, theta functions and Euler products. Differential equations associated with theta functions are solved to reveal these recurrence relations. Interesting properties are investigated including an analogue of Clausen's identity, asymptotic behaviour of the sequences and finally two conjectures for congruence properties are given.