New representations for σ(q) via reciprocity theorems

Two new representations for Ramanujan’s function σ(q) are obtained. The proof of the first one uses the three-variable reciprocity theorem due to Soon-Yi Kang and a transformation due to R.P. Agarwal while that of the second uses the four-variable reciprocity theorem due to George Andrews and a generalization of a recent transformation of Andrews, Schultz, Yee, and the second author. The advantage of these representations is that they involve free complex parameters—one in the first representation, and two in the second. In the course of obtaining these results, we arrive at one- and two-variable generalizations of σ(q).