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ExactPolygamma(PeterO.Numbers.EInteger,PeterO.Numbers.ERational)

Method (no overloads)

Summary

psi^(n)(q) exactly, for an odd order n and a whole or half-whole q other than a pole,
or null: a rational times pi^(n + 1), plus a rational.

Remarks

psi^(n)(1) = n! zeta(n + 1) and psi^(n)(1/2) = (2^(n + 1) - 1) n! zeta(n + 1) for an odd
n (DLMF 5.15.2, 5.15.3), where zeta(2m) = (-1)^(m + 1) B_(2m) (2 pi)^(2m)/(2 (2m)!) is a
rational times pi^(2m); the recurrence psi^(n)(z + 1) = psi^(n)(z) - n!/z^(n + 1) moves
the argument to either. An even order wants zeta at an odd number, or Euler's constant, which
have no closed form here. https://github.com/asc-community/AngouriMath/issues/171

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ExactPolygamma​(PeterO.​Numbers.​EInteger,​PeterO.​Numbers.​ERational)

ExactPolygamma(PeterO.Numbers.EInteger,PeterO.Numbers.ERational)