The functions , , are called the polygamma functions. In particular, is the trigamma function; , , are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. This includes asymptotic expansions: compare §§2.1(ii)–2.1(iii).
For see §24.2(i).
For continued fractions for and see Cuyt et al. (2008, pp. 231–238).