TY - JOUR

T1 - ON THE PARITY OF THE GENERALISED FROBENIUS PARTITION FUNCTIONS

AU - Andrews, GEORGE E.

AU - Sellers, JAMES A.

AU - Soufan, FARES

N1 - Funding Information:
The first author was partially supported by Simons Foundation Grant 633284.
Publisher Copyright:
©

PY - 2022/12/15

Y1 - 2022/12/15

N2 - Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, and enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter Our goal is to identify an infinite family of values of k such that is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers all primes and all values <![CDATA[ $r, 0 < r such that is a quadratic nonresidue modulo for all 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 is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

AB - Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, and enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter Our goal is to identify an infinite family of values of k such that is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers all primes and all values <![CDATA[ $r, 0 < r such that is a quadratic nonresidue modulo for all 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 is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

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U2 - 10.1017/s0004972722000594

DO - 10.1017/s0004972722000594

M3 - Article

AN - SCOPUS:85142503518

SN - 0004-9727

VL - 106

SP - 431

EP - 436

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

IS - 3

ER -