On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$
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arXiv
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| Format: | Preprint |
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2022
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| author | Andrews, George E. Sellers, James A. Soufan, Fares |
| author_facet | Andrews, George E. Sellers, James A. Soufan, Fares |
| contents | In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $ϕ_k(n)$ and $cϕ_k(n),$ 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 $ϕ_k(n)$ 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 $p,$
$$
ϕ_{p\ell-1}(pn+r) \equiv 0 \pmod{2}
$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_00215 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$ Andrews, George E. Sellers, James A. Soufan, Fares Number Theory Combinatorics 11P83, 05A17 In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $ϕ_k(n)$ and $cϕ_k(n),$ 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 $ϕ_k(n)$ 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 $p,$ $$ ϕ_{p\ell-1}(pn+r) \equiv 0 \pmod{2} $$ 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. |
| title | On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$ |
| topic | Number Theory Combinatorics 11P83, 05A17 |
| url | https://arxiv.org/abs/2201.00215 |