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The question is asked in the mathstackexchange. There is no answer. I can't solve it by hand, also. Maybe we can solve it with Mathematica.
enter image description here

Appreciate all your help, effort.

Best regards.

  \[Psi][0][τ]:=  Piecewise[{{0,  t<0}, { 1-t/h}}];
      \[Psi][n][τ]:=  Piecewise[{{0,  t < T-h, {1-(T-t)/h}}];
         \[Psi][i][τ]:=   Piecewise[{(t-(i-h)*h)/h],(i-1)*h<=t<=i*h},{((i+1)*h-t)/h,i*h<=t<=(i+1)*h }}];
    
        ((ρ+1)^(1-α)/(Gamma[α]))*Integrate[(z^(ρ+1)-τ^(ρ+1))^(α-1)*τ^ρ*\[Psi][i][τ], τ];
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    $\begingroup$ What have you tried using Mathematica? Please post any code, so that we can copy and paste it. $\endgroup$ – mgamer Jan 19 at 7:42
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    $\begingroup$ The actual antiderivatives involved here are Integrate[(z^(ρ+1)-τ^(ρ+1))^(α-1)*τ^ρ, τ] (simple) and Integrate[(z^(ρ+1)-τ^(ρ+1))^(α-1)*τ^ρ*τ, τ] (hypergeometric), which can both be done by Mathematica. The tricky bit is the bookkeeping of coefficients and indices $i,j$. $\endgroup$ – Roman Jan 19 at 8:20

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