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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,
5.15.2 ψ(n)(1)=(1)n+1n!ζ(n+1),
5.15.3 ψ(n)(12)=(1)n+1n!(2n+11)ζ(n+1),
5.15.4 ψ(n12)=12π24k=1n11(2k1)2,
5.15.5 ψ(n)(z+1)=ψ(n)(z)+(1)nn!zn1,
5.15.6 ψ(n)(1z)+(1)n1ψ(n)(z)=(1)nπdndzncot(πz),
5.15.7 ψ(n)(mz)=1mn+1k=0m1ψ(n)(z+km).

As z in |phz|πδ

5.15.8 ψ(z)1z+12z2+k=1B2kz2k+1,
5.15.9 ψ(n)(z)(1)n1((n1)!zn+n!2zn+1+k=1(2k+n1)!(2k)!B2kz2k+n).

For B2k see §24.2(i).

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