Existence and linear independence theorem for linear fractional differential equations with constant coefficients

Abstract We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ {l\in\mathbb{N}} is the ceiling for the highest derivative of order α, l - 1 < α ≤ l {l-1<\alpha\leq l} . If β i < α {\beta_{i}<\alpha} are the other derivatives, the existing theory requires α - max ⁡ { β i } ≥ l - 1 {\alpha-\max\{\beta_{i}\}\geq l-1} for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann–Liouville or Caputo fractional derivatives.