Transmutation operators intertwining first-order and distributed-order derivatives

We construct transmutations that intertwine the operator of fractional differentiation of distributed order and the first-order derivative. Fractional differentiation is considered in two forms, in the sense of Riemann-Liouville and in the sense of Gerasimov-Caputo. For each of them, a corresponding transmutation operator is constructed. Distributed differentiation is given using a Lebesgue-Stieltjes measure. This covers both continuously and discretely distributed order fractional operators, as well as their combinations. In the case when the measure is concentrated at a point, the found transmutation operators coincide with the Stankovich transforms. Therefore, the constructed operators, as well as a special function arising at that, are generalizations of the Stankovich transforms and the Wright function to the case of distributed parameters. The found transmutations make it possible to find solutions for evolutionary equations of distributed order in terms of the corresponding first-order equations.