Integral and series representations of the digamma and polygamma functions

We obtain a variety of series and integral representations of the digamma function $ψ(a)$. These in turn provide representations of the evaluations $ψ(p/q)$ at rational argument and for the polygamma function $ψ^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $γ_0(a)=-ψ(a)$. Several other results are obtained, including product representations for $\exp[γ_0(a)]$ and for the Gamma function $Γ(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $ψ(a)$ and the Euler constant $γ=-ψ(1)$.

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