To the properties of one Fox function

The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.