<p>The aim of this paper is to contribute to the development of analytic tools to study fractional operators and equations, seeking unification in the theory and application of distinct areas of mathematics in fractional calculus. On the one hand, we derive the closed-form solution of a tridiagonal system of fractional differential equations formulated with general parameters and fractional orders in terms of Riemann–Liouville derivatives, which renders the typical Caputo system as a particular case. A special case is a fractional model where there is conservation of mass and production-destruction of compartments, generalizing the multi-order Bateman equations of radioactive decay in a linear chain. The Mikusiński algebraic formalism is used, which only assumes continuous inputs in contrast to the customary Laplace transform technique, and is combined with symbolic computation. In the solution, a multivariate Mittag-Leffler function is the basis, and it can even be employed when stochastic forcing terms appear. The methodology also gives new formulae for the one-parameter Mittag-Leffler evaluation of a matrix. On the other hand, we deal with fractional operators and associated bounds employing probability theory and normalizations, in the context of synchronous and convex functions, obtaining known results for the Riemann-Liouville, Caputo, and Prabhakar models as a consequence in a simple way. For this task, we view fractional operators as a redistribution of the past through a probability law, providing alternative arguments to previous techniques.</p>