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The present work deal with some spectral properties of the problem (P)        D α b,− (p(x) D α a,+ y)(x) + λ q(x) y(x) = 0, 1 2 < α < 1, a < x < b lim x−→a > (x − a) 1−α y(x) = 0 = y(b) where p, q ∈ C([a, b]) and p(x) > 0, q(x) > 0, ∀x ∈ [a, b]. D α b,− and D α a,+ are the right-sided and left-sided Riemann-Liouville fractional derivatives of order α ∈ (0, 1), respectively. λ is a scalar parameter. First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space L 2 q [a, b]. Then, we investigate some asymptotic properties of the spectrum as α −→ < 1. We give, in particular, the asymptotic expansion of the first eigenvalue.

The Bessel-Muirhead hypergeometric system (or 0 F 1 -system) in two variables (and three variables) is solved using symmetric series, with an explicit formula for coefficients, in order to express the K-Bessel function as a linear combination of the J-solutions. Limits of this method and suggestions for generalizations to a higher rank are discussed.

First of all, in this paper, we prove the convergence of the nabla hsum to the Riemann-Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Secondly, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case.