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Some new modular relations for the cubic functions. (English) Zbl 1289.33015

Summary: We establish certain relations for the cubic functions \[ \begin{aligned} &S(q): = \sum_{n=0}^{\infty}\frac{(-q; q^2)_nq^{n^2+2n}}{(q^4; q^4)_n}, \\&T(q): =\sum_{n=0}^{\infty}\frac{q^{n^2}}{(q^2; q^2)_n}, \end{aligned} \] which are analogous to Ramanujan’s forty identities for the Rogers-Ramanujan functions. From the relations mentioned above, we deduce some interesting color partition identities.

MSC:

33E20 Other functions defined by series and integrals
11F27 Theta series; Weil representation; theta correspondences
11P82 Analytic theory of partitions
11P84 Partition identities; identities of Rogers-Ramanujan type
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