Lambert Series - Current Usage

Current Usage

In the literature we find Lambert series applied to a wide variety of sums. For example, since is a polylogarithm function, we may refer to any sum of the form

as a Lambert series, assuming that the parameters are suitably restricted. Thus

12\left(\sum_{n=1}^{\infty} n^2 \, \mathrm{Li}_{-1}(q^n)\right)^{\!2} = \sum_{n=1}^{\infty} n^2 \,\mathrm{Li}_{-5}(q^n) - \sum_{n=1}^{\infty} n^4 \, \mathrm{Li}_{-3}(q^n),

which holds for all complex q not on the unit circle, would be considered a Lambert series identity. This identity follows in a straightforward fashion from some identities published by the Indian mathematician S. Ramanujan. A very thorough exploration of Ramanujan's works can be found in the works by Bruce Berndt.

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