is defined as the Hurwitz^4 Zeta function and satisfies ζ(n,0) = ζ(n),the Zeta function.
The function
ψn(z) = (−1)n+1n!ζ(n+ 1 , z) = dn+1
dn+1ln[Γ(z)] (12.140)is referred to as the polygamma function of order n. Observe that
ψ 0 (z) = d
dzlnΓ(z) = Γ′(z)
Γ(z)and ψn(z) = d
n
dznψ 0 (z)The polygamma function of order zero ψ 0 (x)is called the digamma function and is
illustrated in the figure 12-7.
Figure 12-7. The polygamma function of order zero.
(^4) Adolf Hurwitz (1859-1919) German professor of mathematics.