Some new infinite families of congruences modulo 3 for overpartitions into odd parts
Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 255-266
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
Let $ \bar{p}_o(n)$ denote the number of
overpartitions of $n$ in which only odd parts are used. Some
congruences modulo 3 and powers of 2
for the function $ \bar{p}_o(n)$
have been
derived by Hirschhorn and Sellers,
and Lovejoy and Osburn.
In this paper, employing 2-dissections of certain quotients
of theta functions due to Ramanujan,
we prove some new infinite families of Ramanujan-type congruences
for $ \bar{p}_o(n)$
modulo 3. For example, we prove that for $n,
\alpha\geq 0 $,
$$
\bar{p}_o(4^\alpha(24n+17)) \equiv \bar{p}_o(4^\alpha(24n+23)) \equiv 0
\ ({\rm mod}\ 3).
$$
Keywords:
bar denote number overpartitions which only odd parts congruences modulo powers function bar have derived hirschhorn sellers lovejoy osburn paper employing dissections certain quotients theta functions due ramanujan prove infinite families ramanujan type congruences bar modulo example prove alpha geq bar alpha equiv bar alpha equiv mod
Affiliations des auteurs :
Ernest X. W. Xia 1
@article{10_4064_cm142_2_6,
author = {Ernest X. W. Xia},
title = {Some new infinite families of congruences modulo 3 for overpartitions into odd parts},
journal = {Colloquium Mathematicum},
pages = {255--266},
publisher = {mathdoc},
volume = {142},
number = {2},
year = {2016},
doi = {10.4064/cm142-2-6},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm142-2-6/}
}
TY - JOUR AU - Ernest X. W. Xia TI - Some new infinite families of congruences modulo 3 for overpartitions into odd parts JO - Colloquium Mathematicum PY - 2016 SP - 255 EP - 266 VL - 142 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/cm142-2-6/ DO - 10.4064/cm142-2-6 LA - en ID - 10_4064_cm142_2_6 ER -
%0 Journal Article %A Ernest X. W. Xia %T Some new infinite families of congruences modulo 3 for overpartitions into odd parts %J Colloquium Mathematicum %D 2016 %P 255-266 %V 142 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4064/cm142-2-6/ %R 10.4064/cm142-2-6 %G en %F 10_4064_cm142_2_6
Ernest X. W. Xia. Some new infinite families of congruences modulo 3 for overpartitions into odd parts. Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 255-266. doi: 10.4064/cm142-2-6
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