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5 Gamma FunctionProperties

§5.15 Polygamma Functions

The functions ψ(n)(z), n=1,2,, are called the polygamma functions. In particular, ψ(z) is the trigamma function; ψ′′, ψ(3), ψ(4) 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) n,m=1,2,3,, and for ζ(n+1) see §25.6(i).

5.15.1 ψ(z)=k=01(k+z)2,
z0,-1,-2,,
5.15.2 ψ(n)(1)=(-1)n+1n!ζ(n+1),
5.15.3 ψ(n)(12)=(-1)n+1n!(2n+1-1)ζ(n+1),
5.15.4 ψ(n-12)=12π2-4k=1n-11(2k-1)2,
5.15.5 ψ(n)(z+1)=ψ(n)(z)+(-1)nn!z-n-1,
5.15.6 ψ(n)(1-z)+(-1)n-1ψ(n)(z)=(-1)nπdndzncot(πz),
5.15.7 ψ(n)(mz)=1mn+1k=0m-1ψ(n)(z+km).

As z in |phz|π-δ(<π)

5.15.8 ψ(z)1z+12z2+k=1B2kz2k+1.

For B2k see §24.2(i).

For continued fractions for ψ(z) and ψ′′(z) see Cuyt et al. (2008, pp. 231–238).