ON THE PARITY OF THE GENERALISED FROBENIUS PARTITION FUNCTIONS $\boldsymbol {\phi _k(n)}
Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types of combinatorial objects which he called generalised Frobenius...
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Veröffentlicht in: | Bulletin of the Australian Mathematical Society 2022-12, Vol.106 (3), p.431-436 |
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Sprache: | eng |
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Zusammenfassung: | Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions,
$\phi _k(n)$
and
$c\phi _k(n),$
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
$k.$
Our goal is to identify an infinite family of values of k such that
$\phi _k(n)$
is even for all n in a specific arithmetic progression; in particular, we 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
$p,$
$$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$
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 generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question. |
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ISSN: | 0004-9727 1755-1633 |
DOI: | 10.1017/S0004972722000594 |