In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gamma function:
ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z ) . {\displaystyle \psi ^{(m)}(z):={\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\psi (z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z).}
Thus
ψ ( 0 ) ( z ) = ψ ( z ) = Γ ′ ( z ) Γ ( z ) {\displaystyle \psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}}
holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on C ∖ Z ≤ 0 {\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}} . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function.
Integral representation
When m > 0 and Re z > 0, the polygamma function equals
ψ ( m ) ( z ) = ( − 1 ) m + 1 ∫ 0 ∞ t m e − z t 1 − e − t d t = − ∫ 0 1 t z − 1 1 − t ( ln t ) m d t = ( − 1 ) m + 1 m ! ζ ( m + 1 , z ) {\displaystyle {\begin{aligned}\psi ^{(m)}(z)&=(-1)^{m+1}\int _{0}^{\infty }{\frac {t^{m}e^{-zt}}{1-e^{-t}}}\,\mathrm {d} t\\&=-\int _{0}^{1}{\frac {t^{z-1}}{1-t}}(\ln t)^{m}\,\mathrm {d} t\\&=(-1)^{m+1}m!\zeta (m+1,z)\end{aligned}}}
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