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).
In (5.15.2)–(5.15.7) , and for see §25.6(i).
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5.15.2 | |||
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5.15.3 | |||
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5.15.4 | |||
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5.15.5 | |||
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5.15.6 | |||
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5.15.7 | |||
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As in
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5.15.9 | |||
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For see §24.2(i).
For continued fractions for and see Cuyt et al. (2008, pp. 231–238).