Narrow Results

Relations between doubled distinct and self-conjugate t-cores for some small values of t are derived. By employing Ramanujan's simple theta function identities, several infinite families of arithmetic identities for doubled distinct t-cores for t = 3, 4, …, 10 are found.

Let $B_3(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for $B_3(n)$. Moreover, let $\omega (n)$ denote the number of representations of a non-negative integer $n$ in the form $x_1^2+x_2^2+x_3^2+3y_1^2+3y_2^2+3y_3^2$ with $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{Z}.$ We find three arithmetic relations between $B_3(n)$ and $\omega (n)$, such as $\omega (6n+5)=4B_3(6n+4)$.

For prime levels 5 ≤ p ≤ 19, sets of theta quotients are constructed that generate graded rings of modular forms of positive integer weight for Γ₁(p). Action by Γ₀(p) is shown to cyclically permute the generators. This induces symmetric representations for modular forms. The generators are used to deduce representations for the number of t-core partitions of an integer as convolutions of L-functions. Coupled systems of differential equations of level p are constructed for each basis analogous to Ramanujan’s differential equations for Eisenstein series on the full modular group.