Sum[Binomial[a, n] DifferenceDelta[f[x], {x, n}], {n, 0, Infinity}]

Block[{a = 1, f = Exp},
 Sum[Binomial[a, n] DifferenceDelta[f[x], {x, n}], {n, 0, Infinity}]
 ]
(*  E^(1 + x)  *)

Block[{a = 2, f = TrigToExp@*Sin},
  Sum[Binomial[a, n] DifferenceDelta[f[x], {x, n}], {n, 0, Infinity}]
  ] // FullSimplify
(*  Sin[2 + x]  *)