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Polygamma functions of negative order. (English) Zbl 0936.33001

For positive integer n the polygamma function ψ (n) (z) is defined to be the derivative of order n+1 of logΓ(z). The definition can be extended to negative integer n by Liouville’s fractional integration, which gives

ψ -n) (z)=1 (n-1)! 0 z (z-t) n-2 logΓ(t)dt·

The author replaces logΓ(t) by a series representation and integrates term by term to express n!ψ (-n) (z) as an explicit polynomial in z plus a term nζ ' (1-n,z) where R(z)>0 and ζ ' (1-n,z) is the derivative with respect to s of the Hurwitz zeta-function ζ(s,z) evaluated at s=1-n. For example,

ψ (-2) (z)=1 2z(1-z)+1 2zlog(2π)-ζ ' (-1)+ζ ' (-1,z),

where ζ(s)=ζ(s,1) is the Riemann zeta-function.

33B15Gamma, beta and polygamma functions
11M06ζ(s) and L(s,χ)