g = InverseFunction[f];

D[g[y], y]

D[g[y], {y, 2}]

D[g[y], {y, 3}]

f[x_] := 
Log[x] - PolyGamma[x]

Series[
            g[Log[ϵ (x - α β) + α β] - ϵ (y - Log[β] - PolyGamma[α]) - Log[β] - 
                PolyGamma[α]], {ϵ, 0, 2}] // PowerExpand // 
        Normal;

% /. InverseFunction[f][_] -> α /. 
            x -> X + α β /. 
          y -> 
            Y + Log[β] + PolyGamma[0, α] /. ϵ -> 
          1;

Collect[%, {X, Y}, Simplify] /. 
      PolyGamma[i_, α] -> 
        D[Ψ[α], {α, i}]

Series[(ϵ (x - α β) + α β)/g[
                Log[ϵ (x - α β) + α β] - ϵ (y - Log[β] - PolyGamma[α]) - 
                  Log[β] - PolyGamma[α]], {ϵ, 0, 2}] // 
          PowerExpand // Normal;

% /. InverseFunction[f][_] -> α /. 
            x -> X + α β /. 
          y -> 
            Y + Log[β] + PolyGamma[0, α] /. ϵ -> 
          1;

Collect[%, {X, Y}, Simplify] /. 
      PolyGamma[i_, α] -> 
        D[Ψ[α], {α, i}]