Holly Swisher

Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions

In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of “hypergeometric” algebraic varieties that are higher dimensional analogues of Legendre curves.

Inequalities for positive rank and crank moments of overpartitions

In recent work, Andrews, Chan, and Kim extend a result of Garvan about even rank and crank moments of partitions to positive moments. In a similar fashion we extend a result of Mao about even rank moments of overpartitions. We investigate positive Dyson-rank, M2-rank, first residual crank, and second residual crank moments of overpartitions. In particular, we prove a conjecture of Mao which states that the positive Dyson-rank moments are larger than the positive M2-rank moments. We also prove some additional inequalities involving rank and crank moments of overpartitions, including an interlacing property.

Zeros of some level 2 Eisenstein series

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Γ(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allows us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series.

Generalized Legendre curves and quaternionic multiplication

Abstract This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of Q contains a quaternion algebra.

On a conjecture of koike on identities between thompson series and rogers-ramanujan functions

One of the many amazing things Ramanujan did in his lifetime was to list 40 identities involving what are now called the Rogers-Ramanujan functions G(q) and H(q) on one side, and products of functions of the form Q m = Π∞ n=1 (1-q mn ) on the other side. The identities are rather complicated and seem too difficult to guess. Recently however, Koike devised a strategy for finding (but not proving) these types of identities by connecting them to Thompson series. He was able to conjecture many new Rogers-Ramanujan type identities between G(q) and H(q), and Thompson series. Here we prove these identities.

A proof of some conjectures of Mao on partition rank inequalities

Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo 10, and additional results modulo 6 and 10 for the

Mock modularity of the M d -rank of overpartitions

M_2

A NOTE ON THE TRANSCENDENCE OF ZEROS OF A CERTAIN FAMILY OF WEAKLY HOLOMORPHIC FORMS

M2 rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao’s rank inequality conjectures for both the rank and the