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Finding Modular Functions for Ramanujan-Type Identities
Ist Teil von
Annals of combinatorics, 2019-11, Vol.23 (3-4), p.613-657
Ort / Verlag
Cham: Springer International Publishing
Erscheinungsjahr
2019
Link zum Volltext
Quelle
Alma/SFX Local Collection
Beschreibungen/Notizen
This paper is concerned with a class of partition functions
a
(
n
) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for
a
(
m
n
+
t
)
. While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for
p
(
11
n
+
6
)
with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions
p
¯
(
5
n
+
2
)
and
p
¯
(
5
n
+
3
)
and Andrews–Paule’s broken 2-diamond partition functions
▵
2
(
25
n
+
14
)
and
▵
2
(
25
n
+
24
)
. It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews’ singular overpartition functions
Q
¯
3
,
1
(
9
n
+
3
)
and
Q
¯
3
,
1
(
9
n
+
6
)
due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.