Expressions involving Psi are combined as follows

where Q is a rational function, P a univariate polynomial and m and n are rationals.

This is done by applying the following recurrence and reflection formulae.

You can enter the command combine/Psi using either the 1-D or 2-D calling sequence. For example, combine(Psi(x) + Psi(1+x), Psi) is equivalent to $\mathrm{combine}\left(\mathrm{\Psi }\left(x\right)+\mathrm{\Psi }\left(1+x\right)\,\mathrm{\Psi }\right)$.

$\mathrm{combine}\left(\mathrm{\Psi }\left(x\right)+\mathrm{\Psi }\left(1+x\right)\,\mathrm{\Psi }\right)$

$2\mathrm{\Psi }\left(x\right)+\frac{1}{x}$

$\mathrm{combine}\left(\mathrm{\Psi }\left(1-x\right)-\mathrm{\Psi }\left(x\right)+\mathrm{\Psi }\left(y\right)\,\mathrm{\Psi }\right)$

$\mathrm{\pi }\cot \left(\mathrm{\pi }x\right)+\mathrm{\Psi }\left(y\right)$

$\mathrm{combine}\left(\mathrm{\Psi }\left(2\,x-\frac{1}{2}\right)-\mathrm{\Psi }\left(2\,x+\frac{1}{2}\right)\,\mathrm{\Psi }\right)$

$-\frac{2}{{\left(x-\frac{1}{2}\right)}^3}$

$\mathrm{combine}\left(\mathrm{\Psi }\left(1\,x-\frac{1}{2}\right)-\mathrm{\Psi }\left(1\,x+\frac{1}{2}\right)\,\mathrm{\Psi }\right)$

$\frac{1}{{\left(x-\frac{1}{2}\right)}^2}$