A k-Fractional Calculus with Mittag-Leffler Resolvents: Existence Theory and Closed-Form Contraction Thresholds for Singular Boundary-Value Problems

We develop a unified calculus of fractional differential equations built on the k-deformed Gamma function Γ_k, k > 0, in which the classical theory (k = 1), the ordinary exponential (α = k) and the half-order case (α = k/2) appear as special members of one family. We define a k-Riemann-Liouville integral, a k-Caputo derivative and a k-Laplace transform that diagonalizes them, and we show that the two-parameter k-Mittag-Leffler function E^k_{α,β} is the exact resolvent of the linear problem. This gives a variation-of-constants formula and closed forms in terms of exponentials, hyperbolic functions and e^(w²)erfc(−w). For nonlinear initial-value problems we prove global existence and uniqueness by the Banach principle in a Bielecki-type norm weighted by E^k_{α,k}, and existence under sublinear or small linear growth by Schauder's theorem. For a singular Dirichlet problem of order α ∈ (k, 2k] we show that the Green's function changes sign when α < 2k, so cone methods do not apply. We then derive k-analogues of the Euler reflection and Legendre duplication formulas and use them to evaluate the contraction thresholds in closed form. The reflection-case threshold is symmetric under γ ↦ 1−γ and maximal at γ = 1/2. All objects are exact rescalings of classical order-α/k objects, so the contribution is a unified parametrization with explicit k-dependent constants rather than a new function class. The identities and convergence claims are validated numerically.