Editing Generating function (section)

===Lambert series===
{{main article|Lambert series}}

The ''Lambert series'' of a sequence {{math|''a''<sub>''n''</sub>}} is
<math display="block">\operatorname{LG}(a_n;x)=\sum _{n=1}^\infty a_n \frac{x^n}{1-x^n}.</math>Note that in a Lambert series the index {{mvar|n}} starts at 1, not at 0, as the first term would otherwise be undefined. 

The Lambert series coefficients in the power series expansions
<math display="block">b_n := [x^n] \operatorname{LG}(a_n;x)</math>for integers {{math|''n'' ≥ 1}} are related by the [[Divisor sum identities|divisor sum]]
<math display="block">b_n = \sum_{d|n} a_d.</math>The [[Lambert series|main article]] provides several more classical, or at least well-known examples related to special [[arithmetic functions]] in [[number theory]]. As an example of a Lambert series identity not given in the main article, we can show that for {{math|{{abs|''x''}}, {{abs|''xq''}} < 1}} we have that <ref>{{cite web |date=2017 |title=Lambert series identity |url=https://mathoverflow.net/q/140418 |website=Math Overflow}}</ref><math display="block">\sum_{n = 1}^\infty \frac{q^n x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{q^n x^{n^2}}{1-q x^n} + \sum_{n = 1}^\infty \frac{q^n x^{n(n+1)}}{1-x^n}, </math>

where we have the special case identity for the generating function of the [[divisor function]], {{math|''d''(''n'') ≡ ''σ''<sub>0</sub>(''n'')}}, given by<math display="block">\sum_{n = 1}^\infty \frac{x^n}{1-x^n} = \sum_{n = 1}^\infty \frac{x^{n^2} \left(1+x^n\right)}{1-x^n}. </math>