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Using a theorem of Frobenius filtered through partition generating function techniques, we prove partition-theoretic and $q$-series Abelian theorems, yielding analogues of Abels convergence theorem for complex power series, and related formulas. As an application we give a limiting formula for the $q$-bracket of Bloch and Okounkov, an operator from statistical physics connected to the theory of modular forms, as $qto 1$ from within the unit disk.
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $mathbb N$ as limiting values of $q$-series as $qto zeta$ a root of un
We examine partition zeta functions analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over
One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was strengthened by Subba
We identify a class of semi-modular forms invariant on special subgroups of $GL_2(mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like
A $Gamma$-magic rectangle set $MRS_{Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(atimes b)$ whose entries are elements of group $Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $omegain Gamma$