Some integral transformations of a Fox function with four parameters

The study examines the Fox function with four parameters, which arises in the theory of degenerate differential equations with partial derivatives of fractional order. In terms of this function, explicit solutions to the first and second boundary value problems in a half-space were previously derived for the equation with the Bessel operator acting on the spatial variable and a fractional derivative with respect to time.
For the function under consideration, when two of the four parameters are dependent, a Laplace transform formula has been obtained, expressed in terms of the special MacDonald function. Additionally, integral transformation formulas have been derived, expressed through the generalized Wright function and the more general $H$-function of Fox.
An auxiliary tool for proving the obtained formulas is the Mellin–Barnes integral, which is used to express the special function under consideration. The convergence of the improper integrals follows from the asymptotic estimates also provided in the work.
It is shown that for specific values from the Laplace transform formula, known transformation formulas for the exponential function and the Wright function with power multipliers follow.