(PolyGamma[1,1-x/a] – PolyGamma[1,x/a])/(4 a^2)

where PolyGamma[n,x] is defined to be the (n+1)th derivative of Log[Gamma[x]], which is related to the shifted Zeta function (Hurwitz Zeta function).

The series can be quickly matched using the definition in http://mathworld.wolfram.com/PolygammaFunction.html

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