On the Parity of the Generalized Frobenius Partition Functions $\phi_k(n)$

TL;DR

This paper proves an infinite family of parity results for generalized Frobenius partition functions, showing that for certain parameters, these functions are even across specific arithmetic progressions, using elementary methods.

Contribution

It identifies an infinite family of parameters where the generalized Frobenius partition functions are even, expanding understanding of their arithmetic properties with elementary proofs.

Findings

Proves that rac{p ext{l}-1}{p} \, ext{Frobenius partitions are even for specific arithmetic progressions.

Establishes parity results for infinitely many values of the parameter k.

Uses elementary techniques, avoiding complex q-series manipulations.

Abstract

In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $\phi_k(n)$ and $c\phi_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of that Memoir, Andrews proved a number of Ramanujan--like congruences satisfied by specific functions within these two families. In the years that followed, numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ In this brief note, our goal is to identify an {\bf infinite} family of values of $k$ such that $\phi_k(n)$ is even for all $n$ in a specific arithmetic progression; in particular, our primary goal in this work is to prove that, for all positive integers $\ell,$ all primes $p\geq 5,$ and all values $r,$ $0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews' Memoir, classical $q$--series results, and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalized Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.