Axioms, Vol. 12, Pages 514: A q-Series Congruence Inspired by Andrews and Ramanujan

Axioms, Vol. 12, Pages 514: A q-Series Congruence Inspired by Andrews and Ramanujan

Axioms doi: 10.3390/axioms12060514

Authors: Mircea Merca

For each s∈{1,3,5}, we consider Rs(n) to be the number of the partitions of n into parts not congruent to 0, ±s(mod12). In recent years, some relations for computing the value of R3(n) were studied. In this paper, we investigate the parity of Rs(n) when s∈{1,5} and derive the following congruence identity: ∑n=1∞(−q;q)n−12(1+qn)qn2(q;q)2n≡∑n=1∞qn2+q3n2(mod2). For each s∈{1,5}, the number of the partitions of n into parts not congruent to 0, ±s(mod12) is connected with two truncated theta series. Some open problems involving R1(n) and R5(n) are introduced in this context.