Editing Catalan's constant (section)

==Relation to special functions==
{{mvar|G}} appears in values of the second [[polygamma function]], also called the [[trigamma function]], at fractional arguments:<ref name=":0" />

<math display="block">\begin{align}
\psi_1 \left(\tfrac14\right) &= \pi^2 + 8G \\
\psi_1 \left(\tfrac34\right) &= \pi^2 - 8G.
\end{align}</math>

[[Simon Plouffe]] gives an infinite collection of identities between the trigamma function, {{pi}}<sup>2</sup> and Catalan's constant; these are expressible as paths on a graph.

Catalan's constant occurs frequently in relation to the [[Clausen function]], the [[inverse tangent integral]], the [[inverse sine integral]], the [[Barnes G-function|Barnes {{mvar|G}}-function]], as well as integrals and series summable in terms of the aforementioned functions.

As a particular example, by first expressing the [[inverse tangent integral]] in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes {{mvar|G}}-function, the following expression is obtained (see [[Clausen function]] for more):

<math display="block">G=4\pi \log\left( \frac{ G\left(\frac{3}{8}\right) G\left(\frac{7}{8}\right) }{ G\left(\frac{1}{8}\right) G\left(\frac{5}{8}\right) } \right) +4 \pi \log \left( \frac{ \Gamma\left(\frac{3}{8}\right) }{ \Gamma\left(\frac{1}{8}\right) } \right) +\frac{\pi}{2} \log \left( \frac{1+\sqrt{2} }{2 \left(2-\sqrt{2}\right)} \right).</math>

If one defines the '''[[Lerch transcendent]]''' {{math|Φ(''z'',''s'',''α'')}} by
<math display="block">\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s},</math>
then
<math display="block"> G = \tfrac{1}{4}\Phi\left(-1, 2, \tfrac{1}{2}\right).</math>