I take the derivative of the a-th order and see that the formula in the notation changes to Log[m]^a.
This works with the rules, but I quite don’t understand why this formula doesn’t simplify explicitly for any positive integer (Propagate and simplify as an “a-th” power of Log[m] without returning formula with derivative)
And is it possible to expand the definition for real a-th fractional derivative?
It seems that a derivative with non-numeric index does not distribute into a sum with non-numeric extrema, I don’t know why. You can force it to, however, for example with a rule:
distrib = D[Sum[f_, l__], v__] :> Sum[D[f, v], l];
D[Sum[m^(k - 1) f[m, a], {m, 1, a}], {k, a}] /. distrib
(*In[]:=*) distrib = HoldPattern[D[Sum[f_, l__], v__]] :> Sum[D[f, v], l];
Hold[D[Sum[m^(k - 1) f[m, n], {m, 1, n}], {k, a}]] /.
distrib // ReleaseHold
(*Out[]:=*) Sum[m^(-1 + k)*f[m, n]*Log[m]^a, {m, 1, n}]
Oh, this variant worked. And also for FractionalD
(*In[]:=*) distribFD =
HoldPattern[FractionalD[Sum[f_, l__], v__]] :>
Sum[FractionalD[f, v], l];
Hold[FractionalD[Sum[m^(k - 1) f[m, n], {m, 1, n}], {k, a}]] /.
distribFD // ReleaseHold
(*Out[]:=*) Sum[-((m^(-1 + k)*f[m, n]*(-1 + GammaRegularized[-a, k*Log[m]])*
(k*Log[m])^a)/k^a), {m, 1, n}]
You are right, I didn’t realize we need HoldPattern because I had already tried this other solution
Unprotect[D];
D[Sum[f_, l__], v__] := Sum[D[f, v], l];
Protect[D];
D[Sum[m^(k - 1) f[m, n], {m, 1, n}], {k, a}]
and forgot to restart the kernel. Unprotecting D seems dangerous.