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.

Read more about this topic:  Lambert Series

Famous quotes containing the words current and/or usage:

    “But there, where I have garnered up my heart,
    Where either I must live or bear no life;
    The fountain from the which my current runs
    Or else dries up: to be discarded thence,
    Or keep it as a cistern for foul toads
    To knot and gender in!”
    —William Shakespeare (1564–1616)

    “I am using it [the word ‘perceive’] here in such a way that to say of an object that it is perceived does not entail saying that it exists in any sense at all. And this is a perfectly correct and familiar usage of the word.”
    —A.J. (Alfred Jules)